____________________________________________________________________________________
Added January 2, 2019.
Problem 2.2.5.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + \left ( a y + b x^k \right ) w_y = 0 \]
Mathematica ✓
ClearAll[w, x, y, k]; pde = D[w[x, y], x] + (a*y + b*x^k)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to c_1\left (a^{-k-1} e^{-a x} \left (b e^{a x} \text {Gamma}(k+1,a x)+y a^{k+1}\right )\right )\right \}\right \} \]
Maple ✓
w:='w';x:='x';y:='y';a:='a';k:='k';b:='b'; pde := diff(w(x,y),x)+ (a*y+b*x^k)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac {{{\rm e}^{-ax}} \left ( {x}^{k}{{\rm e}^{1/2\,ax}} \left ( ax \right ) ^{-k/2} \WhittakerM \left ( k/2,k/2+1/2,ax \right ) b-aky-ya \right ) }{a \left ( k+1 \right ) }} \right ) \]
____________________________________________________________________________________
Added January 2, 2019.
Problem 2.2.5.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + \left ( a x^k y+b x^n \right ) w_y = 0 \]
Mathematica ✓
ClearAll[w, x, y, n, k]; pde = D[w[x, y], x] + (a*x^k*y + b*x^n)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to c_1\left (\frac {a^{-\frac {n}{k+1}-\frac {1}{k+1}} e^{-\frac {a x^{k+1}}{k+1}} \left (b (k+1)^{\frac {n}{k+1}+\frac {1}{k+1}} e^{\frac {a x^{k+1}}{k+1}} \text {Gamma}\left (\frac {n}{k+1}+\frac {1}{k+1},\frac {a x^{k+1}}{k+1}\right )+k y a^{\frac {n}{k+1}+\frac {1}{k+1}}+y a^{\frac {n}{k+1}+\frac {1}{k+1}}\right )}{k+1}\right )\right \}\right \} \]
Maple ✓
w:='w';x:='x';y:='y';a:='a';k:='k';b:='b';n:='n'; pde := diff(w(x,y),x)+ (a*x^k*y+b*x^n)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac {1}{a \left ( 2\,{k}^{2}n+3\,k{n}^{2}+{n}^{3}+2\,{k}^{2}+10\,kn+6\,{n}^{2}+7\,k+11\,n+6 \right ) } \left ( -6\,{{\rm e}^{-{\frac {a{x}^{k+1}}{k+1}}}}ya+{{\rm e}^{-1/2\,{\frac {a{x}^{k+1}}{k+1}}}}{x}^{-k+n} \left ( {\frac {a{x}^{k+1}}{k+1}} \right ) ^{-1/2\,{\frac {k+n+2}{k+1}}} \WhittakerM \left ( 1/2\,{\frac {k+n+2}{k+1}},1/2\,{\frac {2\,k+n+3}{k+1}},{\frac {a{x}^{k+1}}{k+1}} \right ) bk{n}^{2}+2\,{{\rm e}^{-1/2\,{\frac {a{x}^{k+1}}{k+1}}}}{x}^{-k+n} \left ( {\frac {a{x}^{k+1}}{k+1}} \right ) ^{-1/2\,{\frac {k+n+2}{k+1}}} \WhittakerM \left ( -1/2\,{\frac {k-n}{k+1}},1/2\,{\frac {2\,k+n+3}{k+1}},{\frac {a{x}^{k+1}}{k+1}} \right ) bkn+6\,{{\rm e}^{-1/2\,{\frac {a{x}^{k+1}}{k+1}}}}{x}^{-k+n} \left ( {\frac {a{x}^{k+1}}{k+1}} \right ) ^{-1/2\,{\frac {k+n+2}{k+1}}} \WhittakerM \left ( 1/2\,{\frac {k+n+2}{k+1}},1/2\,{\frac {2\,k+n+3}{k+1}},{\frac {a{x}^{k+1}}{k+1}} \right ) bkn+{{\rm e}^{-1/2\,{\frac {a{x}^{k+1}}{k+1}}}}{x}^{n+1} \left ( {\frac {a{x}^{k+1}}{k+1}} \right ) ^{-1/2\,{\frac {k+n+2}{k+1}}} \WhittakerM \left ( -1/2\,{\frac {k-n}{k+1}},1/2\,{\frac {2\,k+n+3}{k+1}},{\frac {a{x}^{k+1}}{k+1}} \right ) ab{k}^{2}+2\,{{\rm e}^{-1/2\,{\frac {a{x}^{k+1}}{k+1}}}}{x}^{n+1} \left ( {\frac {a{x}^{k+1}}{k+1}} \right ) ^{-1/2\,{\frac {k+n+2}{k+1}}} \WhittakerM \left ( -1/2\,{\frac {k-n}{k+1}},1/2\,{\frac {2\,k+n+3}{k+1}},{\frac {a{x}^{k+1}}{k+1}} \right ) abk+{{\rm e}^{-1/2\,{\frac {a{x}^{k+1}}{k+1}}}}{x}^{-k+n} \left ( {\frac {a{x}^{k+1}}{k+1}} \right ) ^{-1/2\,{\frac {k+n+2}{k+1}}} \WhittakerM \left ( -1/2\,{\frac {k-n}{k+1}},1/2\,{\frac {2\,k+n+3}{k+1}},{\frac {a{x}^{k+1}}{k+1}} \right ) b{k}^{2}n+2\,{{\rm e}^{-1/2\,{\frac {a{x}^{k+1}}{k+1}}}}{x}^{-k+n} \left ( {\frac {a{x}^{k+1}}{k+1}} \right ) ^{-1/2\,{\frac {k+n+2}{k+1}}} \WhittakerM \left ( 1/2\,{\frac {k+n+2}{k+1}},1/2\,{\frac {2\,k+n+3}{k+1}},{\frac {a{x}^{k+1}}{k+1}} \right ) b{k}^{2}n+2\,{{\rm e}^{-1/2\,{\frac {a{x}^{k+1}}{k+1}}}}{x}^{-k+n} \left ( {\frac {a{x}^{k+1}}{k+1}} \right ) ^{-1/2\,{\frac {k+n+2}{k+1}}} \WhittakerM \left ( -1/2\,{\frac {k-n}{k+1}},1/2\,{\frac {2\,k+n+3}{k+1}},{\frac {a{x}^{k+1}}{k+1}} \right ) b-{{\rm e}^{-{\frac {a{x}^{k+1}}{k+1}}}}ya{n}^{3}-6\,{{\rm e}^{-{\frac {a{x}^{k+1}}{k+1}}}}ya{n}^{2}-7\,{{\rm e}^{-{\frac {a{x}^{k+1}}{k+1}}}}yak-11\,{{\rm e}^{-{\frac {a{x}^{k+1}}{k+1}}}}yan-2\,{{\rm e}^{-{\frac {a{x}^{k+1}}{k+1}}}}ya{k}^{2}+4\,{{\rm e}^{-1/2\,{\frac {a{x}^{k+1}}{k+1}}}}{x}^{-k+n} \left ( {\frac {a{x}^{k+1}}{k+1}} \right ) ^{-1/2\,{\frac {k+n+2}{k+1}}} \WhittakerM \left ( 1/2\,{\frac {k+n+2}{k+1}},1/2\,{\frac {2\,k+n+3}{k+1}},{\frac {a{x}^{k+1}}{k+1}} \right ) b+5\,{{\rm e}^{-1/2\,{\frac {a{x}^{k+1}}{k+1}}}}{x}^{-k+n} \left ( {\frac {a{x}^{k+1}}{k+1}} \right ) ^{-1/2\,{\frac {k+n+2}{k+1}}} \WhittakerM \left ( 1/2\,{\frac {k+n+2}{k+1}},1/2\,{\frac {2\,k+n+3}{k+1}},{\frac {a{x}^{k+1}}{k+1}} \right ) b{k}^{2}+{{\rm e}^{-1/2\,{\frac {a{x}^{k+1}}{k+1}}}}{x}^{n+1} \left ( {\frac {a{x}^{k+1}}{k+1}} \right ) ^{-1/2\,{\frac {k+n+2}{k+1}}} \WhittakerM \left ( -1/2\,{\frac {k-n}{k+1}},1/2\,{\frac {2\,k+n+3}{k+1}},{\frac {a{x}^{k+1}}{k+1}} \right ) ab+5\,{{\rm e}^{-1/2\,{\frac {a{x}^{k+1}}{k+1}}}}{x}^{-k+n} \left ( {\frac {a{x}^{k+1}}{k+1}} \right ) ^{-1/2\,{\frac {k+n+2}{k+1}}} \WhittakerM \left ( -1/2\,{\frac {k-n}{k+1}},1/2\,{\frac {2\,k+n+3}{k+1}},{\frac {a{x}^{k+1}}{k+1}} \right ) bk+{{\rm e}^{-1/2\,{\frac {a{x}^{k+1}}{k+1}}}}{x}^{-k+n} \left ( {\frac {a{x}^{k+1}}{k+1}} \right ) ^{-1/2\,{\frac {k+n+2}{k+1}}} \WhittakerM \left ( -1/2\,{\frac {k-n}{k+1}},1/2\,{\frac {2\,k+n+3}{k+1}},{\frac {a{x}^{k+1}}{k+1}} \right ) bn+8\,{{\rm e}^{-1/2\,{\frac {a{x}^{k+1}}{k+1}}}}{x}^{-k+n} \left ( {\frac {a{x}^{k+1}}{k+1}} \right ) ^{-1/2\,{\frac {k+n+2}{k+1}}} \WhittakerM \left ( 1/2\,{\frac {k+n+2}{k+1}},1/2\,{\frac {2\,k+n+3}{k+1}},{\frac {a{x}^{k+1}}{k+1}} \right ) bk+4\,{{\rm e}^{-1/2\,{\frac {a{x}^{k+1}}{k+1}}}}{x}^{-k+n} \left ( {\frac {a{x}^{k+1}}{k+1}} \right ) ^{-1/2\,{\frac {k+n+2}{k+1}}} \WhittakerM \left ( 1/2\,{\frac {k+n+2}{k+1}},1/2\,{\frac {2\,k+n+3}{k+1}},{\frac {a{x}^{k+1}}{k+1}} \right ) bn+{{\rm e}^{-1/2\,{\frac {a{x}^{k+1}}{k+1}}}}{x}^{-k+n} \left ( {\frac {a{x}^{k+1}}{k+1}} \right ) ^{-1/2\,{\frac {k+n+2}{k+1}}} \WhittakerM \left ( -1/2\,{\frac {k-n}{k+1}},1/2\,{\frac {2\,k+n+3}{k+1}},{\frac {a{x}^{k+1}}{k+1}} \right ) b{k}^{3}+{{\rm e}^{-1/2\,{\frac {a{x}^{k+1}}{k+1}}}}{x}^{-k+n} \left ( {\frac {a{x}^{k+1}}{k+1}} \right ) ^{-1/2\,{\frac {k+n+2}{k+1}}} \WhittakerM \left ( 1/2\,{\frac {k+n+2}{k+1}},1/2\,{\frac {2\,k+n+3}{k+1}},{\frac {a{x}^{k+1}}{k+1}} \right ) b{k}^{3}+4\,{{\rm e}^{-1/2\,{\frac {a{x}^{k+1}}{k+1}}}}{x}^{-k+n} \left ( {\frac {a{x}^{k+1}}{k+1}} \right ) ^{-1/2\,{\frac {k+n+2}{k+1}}} \WhittakerM \left ( -1/2\,{\frac {k-n}{k+1}},1/2\,{\frac {2\,k+n+3}{k+1}},{\frac {a{x}^{k+1}}{k+1}} \right ) b{k}^{2}+{{\rm e}^{-1/2\,{\frac {a{x}^{k+1}}{k+1}}}}{x}^{-k+n} \left ( {\frac {a{x}^{k+1}}{k+1}} \right ) ^{-1/2\,{\frac {k+n+2}{k+1}}} \WhittakerM \left ( 1/2\,{\frac {k+n+2}{k+1}},1/2\,{\frac {2\,k+n+3}{k+1}},{\frac {a{x}^{k+1}}{k+1}} \right ) b{n}^{2}-10\,{{\rm e}^{-{\frac {a{x}^{k+1}}{k+1}}}}yakn-2\,{{\rm e}^{-{\frac {a{x}^{k+1}}{k+1}}}}ya{k}^{2}n-3\,{{\rm e}^{-{\frac {a{x}^{k+1}}{k+1}}}}yak{n}^{2} \right ) } \right ) \]
____________________________________________________________________________________
Added January 2, 2019.
Problem 2.2.5.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + \left ( a y^2+b x^n \right ) w_y = 0 \]
Mathematica ✓
ClearAll[w, x, y, n, a, b]; pde = D[w[x, y], x] + (a*y^2 + b*x^n)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to c_1\left (\frac {2 \left (-a x y \text {BesselJ}\left (\frac {1}{n+2},\frac {2 \sqrt {a} \sqrt {b} x^{\frac {n+2}{2}}}{n+2}\right )-\sqrt {a} \sqrt {b} x^{\frac {n}{2}+1} \text {BesselJ}\left (\frac {1}{n+2}-1,\frac {2 \sqrt {a} \sqrt {b} x^{\frac {n+2}{2}}}{n+2}\right )\right )}{-\sqrt {a} \sqrt {b} x^{\frac {n}{2}+1} \text {BesselJ}\left (\frac {n+1}{n+2},\frac {2 \sqrt {a} \sqrt {b} x^{\frac {n}{2}+1}}{n+2}\right )+2 a x y \text {BesselJ}\left (-\frac {1}{n+2},\frac {2 \sqrt {a} \sqrt {b} x^{\frac {n+2}{2}}}{n+2}\right )+\text {BesselJ}\left (-\frac {1}{n+2},\frac {2 \sqrt {a} \sqrt {b} x^{\frac {n+2}{2}}}{n+2}\right )+\sqrt {a} \sqrt {b} x^{\frac {n}{2}+1} \text {BesselJ}\left (-\frac {n+3}{n+2},\frac {2 \sqrt {a} \sqrt {b} x^{\frac {n+2}{2}}}{n+2}\right )}\right )\right \}\right \} \]
Maple ✓
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n'; pde := diff(w(x,y),x)+ (a*y^2+b*x^n)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( {1 \left ( -\BesselY \left ( {\frac {n+3}{n+2}},2\,{\frac {\sqrt {ab}{x}^{n/2}x}{n+2}} \right ) \sqrt {ab}{x}^{n/2}x+\BesselY \left ( \left ( n+2 \right ) ^{-1},2\,{\frac {\sqrt {ab}{x}^{n/2}x}{n+2}} \right ) yax+\BesselY \left ( \left ( n+2 \right ) ^{-1},2\,{\frac {\sqrt {ab}{x}^{n/2}x}{n+2}} \right ) \right ) \left ( \BesselJ \left ( {\frac {n+3}{n+2}},2\,{\frac {\sqrt {ab}{x}^{n/2}x}{n+2}} \right ) \sqrt {ab}{x}^{n/2}x-\BesselJ \left ( \left ( n+2 \right ) ^{-1},2\,{\frac {\sqrt {ab}{x}^{n/2}x}{n+2}} \right ) yax-\BesselJ \left ( \left ( n+2 \right ) ^{-1},2\,{\frac {\sqrt {ab}{x}^{n/2}x}{n+2}} \right ) \right ) ^{-1}} \right ) \]
____________________________________________________________________________________
Added January 2, 2019.
Problem 2.2.5.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + \left ( y^2+a n x^{n-1} -a^2 x^{2 n} \right ) w_y = 0 \]
Mathematica ✗
ClearAll[w, x, y, n, a]; pde = D[w[x, y], x] + (y^2 + a*n*x^(n - 1) - a^2*x^(2*n))*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \text {Failed} \]
Maple ✓
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n'; pde := diff(w(x,y),x)+ (y^2+a*n*x^(n-1)-a^2*x^(2*n))*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( 2\,{({x}^{5/2\,n+2}a-{x}^{3/2\,n+2}y){{\rm e}^{-{\frac {{x}^{n+1}a}{n+1}}}} \left ( -4\,yx \WhittakerM \left ( 1/2\,{\frac {n+2}{n+1}},1/2\,{\frac {2\,n+3}{n+1}},-2\,{\frac {{x}^{n+1}a}{n+1}} \right ) -2\,yx \WhittakerM \left ( -1/2\,{\frac {n}{n+1}},1/2\,{\frac {2\,n+3}{n+1}},-2\,{\frac {{x}^{n+1}a}{n+1}} \right ) +2\,a{x}^{n+1} \WhittakerM \left ( -1/2\,{\frac {n}{n+1}},1/2\,{\frac {2\,n+3}{n+1}},-2\,{\frac {{x}^{n+1}a}{n+1}} \right ) -2\, \WhittakerM \left ( -1/2\,{\frac {n}{n+1}},1/2\,{\frac {2\,n+3}{n+1}},-2\,{\frac {{x}^{n+1}a}{n+1}} \right ) {x}^{2\,n+2}{a}^{2}+8\,a{x}^{n+1} \WhittakerM \left ( 1/2\,{\frac {n+2}{n+1}},1/2\,{\frac {2\,n+3}{n+1}},-2\,{\frac {{x}^{n+1}a}{n+1}} \right ) -11\,{n}^{2} \left ( -2\,{\frac {{x}^{n+1}a}{n+1}} \right ) ^{1/2\,{\frac {3\,n+4}{n+1}}}{{\rm e}^{{\frac {{x}^{n+1}a}{n+1}}}}+5\, \WhittakerM \left ( -1/2\,{\frac {n}{n+1}},1/2\,{\frac {2\,n+3}{n+1}},-2\,{\frac {{x}^{n+1}a}{n+1}} \right ) n-20\,n \left ( -2\,{\frac {{x}^{n+1}a}{n+1}} \right ) ^{1/2\,{\frac {3\,n+4}{n+1}}}{{\rm e}^{{\frac {{x}^{n+1}a}{n+1}}}}-2\,{n}^{3} \left ( -2\,{\frac {{x}^{n+1}a}{n+1}} \right ) ^{1/2\,{\frac {3\,n+4}{n+1}}}{{\rm e}^{{\frac {{x}^{n+1}a}{n+1}}}}+3\,an{x}^{n+1} \WhittakerM \left ( -1/2\,{\frac {n}{n+1}},1/2\,{\frac {2\,n+3}{n+1}},-2\,{\frac {{x}^{n+1}a}{n+1}} \right ) -2\, \WhittakerM \left ( -1/2\,{\frac {n}{n+1}},1/2\,{\frac {2\,n+3}{n+1}},-2\,{\frac {{x}^{n+1}a}{n+1}} \right ) {x}^{2\,n+2}{a}^{2}n+2\,a \WhittakerM \left ( -1/2\,{\frac {n}{n+1}},1/2\,{\frac {2\,n+3}{n+1}},-2\,{\frac {{x}^{n+1}a}{n+1}} \right ) y{x}^{n+2}-{n}^{2}yx \WhittakerM \left ( 1/2\,{\frac {n+2}{n+1}},1/2\,{\frac {2\,n+3}{n+1}},-2\,{\frac {{x}^{n+1}a}{n+1}} \right ) -{n}^{2}yx \WhittakerM \left ( -1/2\,{\frac {n}{n+1}},1/2\,{\frac {2\,n+3}{n+1}},-2\,{\frac {{x}^{n+1}a}{n+1}} \right ) -4\,nyx \WhittakerM \left ( 1/2\,{\frac {n+2}{n+1}},1/2\,{\frac {2\,n+3}{n+1}},-2\,{\frac {{x}^{n+1}a}{n+1}} \right ) -3\,nyx \WhittakerM \left ( -1/2\,{\frac {n}{n+1}},1/2\,{\frac {2\,n+3}{n+1}},-2\,{\frac {{x}^{n+1}a}{n+1}} \right ) +a{n}^{2}{x}^{n+1} \WhittakerM \left ( -1/2\,{\frac {n}{n+1}},1/2\,{\frac {2\,n+3}{n+1}},-2\,{\frac {{x}^{n+1}a}{n+1}} \right ) +3\,a{n}^{2}{x}^{n+1} \WhittakerM \left ( 1/2\,{\frac {n+2}{n+1}},1/2\,{\frac {2\,n+3}{n+1}},-2\,{\frac {{x}^{n+1}a}{n+1}} \right ) +10\,an{x}^{n+1} \WhittakerM \left ( 1/2\,{\frac {n+2}{n+1}},1/2\,{\frac {2\,n+3}{n+1}},-2\,{\frac {{x}^{n+1}a}{n+1}} \right ) +{n}^{3} \WhittakerM \left ( 1/2\,{\frac {n+2}{n+1}},1/2\,{\frac {2\,n+3}{n+1}},-2\,{\frac {{x}^{n+1}a}{n+1}} \right ) +{n}^{3} \WhittakerM \left ( -1/2\,{\frac {n}{n+1}},1/2\,{\frac {2\,n+3}{n+1}},-2\,{\frac {{x}^{n+1}a}{n+1}} \right ) +5\,{n}^{2} \WhittakerM \left ( 1/2\,{\frac {n+2}{n+1}},1/2\,{\frac {2\,n+3}{n+1}},-2\,{\frac {{x}^{n+1}a}{n+1}} \right ) +4\,{n}^{2} \WhittakerM \left ( -1/2\,{\frac {n}{n+1}},1/2\,{\frac {2\,n+3}{n+1}},-2\,{\frac {{x}^{n+1}a}{n+1}} \right ) +8\,n \WhittakerM \left ( 1/2\,{\frac {n+2}{n+1}},1/2\,{\frac {2\,n+3}{n+1}},-2\,{\frac {{x}^{n+1}a}{n+1}} \right ) +4\, \WhittakerM \left ( 1/2\,{\frac {n+2}{n+1}},1/2\,{\frac {2\,n+3}{n+1}},-2\,{\frac {{x}^{n+1}a}{n+1}} \right ) +2\, \WhittakerM \left ( -1/2\,{\frac {n}{n+1}},1/2\,{\frac {2\,n+3}{n+1}},-2\,{\frac {{x}^{n+1}a}{n+1}} \right ) +2\,an \WhittakerM \left ( -1/2\,{\frac {n}{n+1}},1/2\,{\frac {2\,n+3}{n+1}},-2\,{\frac {{x}^{n+1}a}{n+1}} \right ) y{x}^{n+2}-12\, \left ( -2\,{\frac {{x}^{n+1}a}{n+1}} \right ) ^{1/2\,{\frac {3\,n+4}{n+1}}}{{\rm e}^{{\frac {{x}^{n+1}a}{n+1}}}} \right ) ^{-1}} \right ) \]
____________________________________________________________________________________
Added January 2, 2019.
Problem 2.2.5.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + \left ( y^2 + a x^n y + a x^{n-1} \right ) w_y = 0 \]
Mathematica ✓
ClearAll[w, x, y, n, a]; pde = D[w[x, y], x] + (y^2 + a*x^n*y + a*x^(n - 1))*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to c_1\left (-\frac {(n+1)^{-\frac {1}{n+1}-1} \left ((-1)^{\frac {1}{n+1}+1} x a^{\frac {1}{n+1}} \text {Gamma}\left (\frac {1}{-n-1},-\frac {a x^{n+1}}{n+1}\right )+(-1)^{\frac {1}{n+1}+1} x^2 y a^{\frac {1}{n+1}} \text {Gamma}\left (\frac {1}{-n-1},-\frac {a x^{n+1}}{n+1}\right )+(n+1)^{\frac {1}{n+1}} e^{\frac {a x^{n+1}}{n+1}}+n (n+1)^{\frac {1}{n+1}} e^{\frac {a x^{n+1}}{n+1}}\right )}{x (x y+1)}\right )\right \}\right \} \]
Maple ✓
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n'; pde := diff(w(x,y),x)+ (y^2+a*x^n*y+a*x^(n-1))*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1}{anx \left ( 2\,nyx+yx+2\,n+1 \right ) } \left ( {{\rm e}^{1/2\,{\frac {{x}^{n+1}a}{n+1}}}}{x}^{-n}y \left ( -{\frac {{x}^{n+1}a}{n+1}} \right ) ^{-1/2\,{\frac {n}{n+1}}} \WhittakerM \left ( -1/2\,{\frac {n+2}{n+1}},1/2\,{\frac {1+2\,n}{n+1}},-{\frac {{x}^{n+1}a}{n+1}} \right ) {n}^{3}+{{\rm e}^{1/2\,{\frac {{x}^{n+1}a}{n+1}}}}{x}^{-n}y \left ( -{\frac {{x}^{n+1}a}{n+1}} \right ) ^{-1/2\,{\frac {n}{n+1}}} \WhittakerM \left ( 1/2\,{\frac {n}{n+1}},1/2\,{\frac {1+2\,n}{n+1}},-{\frac {{x}^{n+1}a}{n+1}} \right ) {n}^{3}+2\,{{\rm e}^{1/2\,{\frac {{x}^{n+1}a}{n+1}}}}{x}^{-n}y \left ( -{\frac {{x}^{n+1}a}{n+1}} \right ) ^{-1/2\,{\frac {n}{n+1}}} \WhittakerM \left ( -1/2\,{\frac {n+2}{n+1}},1/2\,{\frac {1+2\,n}{n+1}},-{\frac {{x}^{n+1}a}{n+1}} \right ) {n}^{2}+{{\rm e}^{1/2\,{\frac {{x}^{n+1}a}{n+1}}}}{x}^{-n}y \left ( -{\frac {{x}^{n+1}a}{n+1}} \right ) ^{-1/2\,{\frac {n}{n+1}}} \WhittakerM \left ( 1/2\,{\frac {n}{n+1}},1/2\,{\frac {1+2\,n}{n+1}},-{\frac {{x}^{n+1}a}{n+1}} \right ) {n}^{2}+{{\rm e}^{1/2\,{\frac {{x}^{n+1}a}{n+1}}}}{x}^{-n-1} \left ( -{\frac {{x}^{n+1}a}{n+1}} \right ) ^{-1/2\,{\frac {n}{n+1}}} \WhittakerM \left ( -1/2\,{\frac {n+2}{n+1}},1/2\,{\frac {1+2\,n}{n+1}},-{\frac {{x}^{n+1}a}{n+1}} \right ) {n}^{3}+{{\rm e}^{1/2\,{\frac {{x}^{n+1}a}{n+1}}}}{x}^{-n-1} \left ( -{\frac {{x}^{n+1}a}{n+1}} \right ) ^{-1/2\,{\frac {n}{n+1}}} \WhittakerM \left ( 1/2\,{\frac {n}{n+1}},1/2\,{\frac {1+2\,n}{n+1}},-{\frac {{x}^{n+1}a}{n+1}} \right ) {n}^{3}-{{\rm e}^{1/2\,{\frac {{x}^{n+1}a}{n+1}}}}y \left ( -{\frac {{x}^{n+1}a}{n+1}} \right ) ^{-1/2\,{\frac {n}{n+1}}} \WhittakerM \left ( -1/2\,{\frac {n+2}{n+1}},1/2\,{\frac {1+2\,n}{n+1}},-{\frac {{x}^{n+1}a}{n+1}} \right ) a{n}^{2}x+{{\rm e}^{1/2\,{\frac {{x}^{n+1}a}{n+1}}}}{x}^{-n}y \left ( -{\frac {{x}^{n+1}a}{n+1}} \right ) ^{-1/2\,{\frac {n}{n+1}}} \WhittakerM \left ( -1/2\,{\frac {n+2}{n+1}},1/2\,{\frac {1+2\,n}{n+1}},-{\frac {{x}^{n+1}a}{n+1}} \right ) n+2\,{{\rm e}^{1/2\,{\frac {{x}^{n+1}a}{n+1}}}}{x}^{-n-1} \left ( -{\frac {{x}^{n+1}a}{n+1}} \right ) ^{-1/2\,{\frac {n}{n+1}}} \WhittakerM \left ( -1/2\,{\frac {n+2}{n+1}},1/2\,{\frac {1+2\,n}{n+1}},-{\frac {{x}^{n+1}a}{n+1}} \right ) {n}^{2}+{{\rm e}^{1/2\,{\frac {{x}^{n+1}a}{n+1}}}}{x}^{-n-1} \left ( -{\frac {{x}^{n+1}a}{n+1}} \right ) ^{-1/2\,{\frac {n}{n+1}}} \WhittakerM \left ( 1/2\,{\frac {n}{n+1}},1/2\,{\frac {1+2\,n}{n+1}},-{\frac {{x}^{n+1}a}{n+1}} \right ) {n}^{2}-2\,{{\rm e}^{1/2\,{\frac {{x}^{n+1}a}{n+1}}}}y \left ( -{\frac {{x}^{n+1}a}{n+1}} \right ) ^{-1/2\,{\frac {n}{n+1}}} \WhittakerM \left ( -1/2\,{\frac {n+2}{n+1}},1/2\,{\frac {1+2\,n}{n+1}},-{\frac {{x}^{n+1}a}{n+1}} \right ) anx+{{\rm e}^{1/2\,{\frac {{x}^{n+1}a}{n+1}}}}{x}^{-n-1} \left ( -{\frac {{x}^{n+1}a}{n+1}} \right ) ^{-1/2\,{\frac {n}{n+1}}} \WhittakerM \left ( -1/2\,{\frac {n+2}{n+1}},1/2\,{\frac {1+2\,n}{n+1}},-{\frac {{x}^{n+1}a}{n+1}} \right ) n-{{\rm e}^{1/2\,{\frac {{x}^{n+1}a}{n+1}}}}y \left ( -{\frac {{x}^{n+1}a}{n+1}} \right ) ^{-1/2\,{\frac {n}{n+1}}} \WhittakerM \left ( -1/2\,{\frac {n+2}{n+1}},1/2\,{\frac {1+2\,n}{n+1}},-{\frac {{x}^{n+1}a}{n+1}} \right ) ax-{{\rm e}^{1/2\,{\frac {{x}^{n+1}a}{n+1}}}} \left ( -{\frac {{x}^{n+1}a}{n+1}} \right ) ^{-1/2\,{\frac {n}{n+1}}} \WhittakerM \left ( -1/2\,{\frac {n+2}{n+1}},1/2\,{\frac {1+2\,n}{n+1}},-{\frac {{x}^{n+1}a}{n+1}} \right ) a{n}^{2}+2\,{{\rm e}^{{\frac {{x}^{n+1}a}{n+1}}}}a{n}^{2}-2\,{{\rm e}^{1/2\,{\frac {{x}^{n+1}a}{n+1}}}} \left ( -{\frac {{x}^{n+1}a}{n+1}} \right ) ^{-1/2\,{\frac {n}{n+1}}} \WhittakerM \left ( -1/2\,{\frac {n+2}{n+1}},1/2\,{\frac {1+2\,n}{n+1}},-{\frac {{x}^{n+1}a}{n+1}} \right ) an+{{\rm e}^{{\frac {{x}^{n+1}a}{n+1}}}}an-{{\rm e}^{1/2\,{\frac {{x}^{n+1}a}{n+1}}}} \left ( -{\frac {{x}^{n+1}a}{n+1}} \right ) ^{-1/2\,{\frac {n}{n+1}}} \WhittakerM \left ( -1/2\,{\frac {n+2}{n+1}},1/2\,{\frac {1+2\,n}{n+1}},-{\frac {{x}^{n+1}a}{n+1}} \right ) a \right ) } \right ) \]
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Added January 2, 2019.
Problem 2.2.5.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + \left ( y^2+a x^n y-a b x^n -b^2 \right ) w_y = 0 \]
Mathematica ✗
ClearAll[w, x, y, n, a, b]; pde = D[w[x, y], x] + (y^2 + a*x^n*y - a*b*x^n - b^2)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \text {\$Aborted} \] Timed out
Maple ✓
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n'; pde := diff(w(x,y),x)+ (y^2+a*x^n*y-a*b*x^n-b^2)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac {1}{-b+y} \left ( \int \!{{\rm e}^{{\frac {x \left ( {x}^{n}a+2\,bn+2\,b \right ) }{n+1}}}}\,{\rm d}xy-\int \!{{\rm e}^{{\frac {x \left ( {x}^{n}a+2\,bn+2\,b \right ) }{n+1}}}}\,{\rm d}xb+{{\rm e}^{{\frac {x \left ( {x}^{n}a+2\,bn+2\,b \right ) }{n+1}}}} \right ) } \right ) \]
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Added January 2, 2019.
Problem 2.2.5.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + \left ( a x^n y^2 + b x^{-n-2} \right ) w_y = 0 \]
Mathematica ✓
ClearAll[w, x, y, n, a, b]; pde = D[w[x, y], x] + (a*x^n*y^2 + b*x^(-n - 2))*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to c_1\left (\frac {-2 a y x^{\frac {1}{2} \sqrt {a} \sqrt {b} \left (\frac {\sqrt {-4 a b+n^2+2 n+1}}{\sqrt {a} \sqrt {b}}-\frac {-n-1}{\sqrt {a} \sqrt {b}}\right )+n+1}-n x^{\frac {1}{2} \sqrt {a} \sqrt {b} \left (\frac {\sqrt {-4 a b+n^2+2 n+1}}{\sqrt {a} \sqrt {b}}-\frac {-n-1}{\sqrt {a} \sqrt {b}}\right )}-\sqrt {-4 a b+n^2+2 n+1} x^{\frac {1}{2} \sqrt {a} \sqrt {b} \left (\frac {\sqrt {-4 a b+n^2+2 n+1}}{\sqrt {a} \sqrt {b}}-\frac {-n-1}{\sqrt {a} \sqrt {b}}\right )}-x^{\frac {1}{2} \sqrt {a} \sqrt {b} \left (\frac {\sqrt {-4 a b+n^2+2 n+1}}{\sqrt {a} \sqrt {b}}-\frac {-n-1}{\sqrt {a} \sqrt {b}}\right )}}{2 a y x^{\frac {1}{2} \sqrt {a} \sqrt {b} \left (-\frac {\sqrt {-4 a b+n^2+2 n+1}}{\sqrt {a} \sqrt {b}}-\frac {-n-1}{\sqrt {a} \sqrt {b}}\right )+n+1}+n x^{\frac {1}{2} \sqrt {a} \sqrt {b} \left (-\frac {\sqrt {-4 a b+n^2+2 n+1}}{\sqrt {a} \sqrt {b}}-\frac {-n-1}{\sqrt {a} \sqrt {b}}\right )}-\sqrt {-4 a b+n^2+2 n+1} x^{\frac {1}{2} \sqrt {a} \sqrt {b} \left (-\frac {\sqrt {-4 a b+n^2+2 n+1}}{\sqrt {a} \sqrt {b}}-\frac {-n-1}{\sqrt {a} \sqrt {b}}\right )}+x^{\frac {1}{2} \sqrt {a} \sqrt {b} \left (-\frac {\sqrt {-4 a b+n^2+2 n+1}}{\sqrt {a} \sqrt {b}}-\frac {-n-1}{\sqrt {a} \sqrt {b}}\right )}}\right )\right \}\right \} \]
Maple ✓
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n'; pde := diff(w(x,y),x)+ (a*x^n*y^2+b*x^(-n-2))*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1}{\sqrt {4\,ab-{n}^{2}-2\,n-1}} \left ( \ln \left ( x \right ) \sqrt {4\,ab-{n}^{2}-2\,n-1}-2\,\arctan \left ( {\frac {2\,a{x}^{n}yx+n+1}{\sqrt {4\,ab-{n}^{2}-2\,n-1}}} \right ) \right ) } \right ) \]
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Added January 2, 2019.
Problem 2.2.5.8 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + \left ( a x^n y^2 + b m x^{m-1} -a b^2 x^{n+2 m} \right ) w_y = 0 \]
Mathematica ✗
ClearAll[w, x, y, n, a, b, m]; pde = D[w[x, y], x] + (a*x^n*y^2 + b*m*x^(m - 1) - a*b^2*x^(n + 2*m))*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \text {Failed} \]
Maple ✓
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m'; pde := diff(w(x,y),x)+ (a*x^n*y^2 + b*m*x^(m-1) -a*b^2*x^(n+2*m))*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( -2\,{a \left ( {x}^{5/2\,m+2\,n+2}b-{x}^{3/2\,m+2\,n+2}y \right ) {{\rm e}^{-{\frac {{x}^{m+n+1}ab}{m+n+1}}}} \left ( 2\,ay{n}^{2}{x}^{n+1} \WhittakerM \left ( -1/2\,{\frac {m}{m+n+1}},1/2\,{\frac {2\,m+3\,n+3}{m+n+1}},-2\,{\frac {{x}^{m+n+1}ab}{m+n+1}} \right ) +8\,ayn{x}^{n+1} \WhittakerM \left ( 1/2\,{\frac {m+2\,n+2}{m+n+1}},1/2\,{\frac {2\,m+3\,n+3}{m+n+1}},-2\,{\frac {{x}^{m+n+1}ab}{m+n+1}} \right ) +4\,ayn{x}^{n+1} \WhittakerM \left ( -1/2\,{\frac {m}{m+n+1}},1/2\,{\frac {2\,m+3\,n+3}{m+n+1}},-2\,{\frac {{x}^{m+n+1}ab}{m+n+1}} \right ) -2\, \WhittakerM \left ( -1/2\,{\frac {m}{m+n+1}},1/2\,{\frac {2\,m+3\,n+3}{m+n+1}},-2\,{\frac {{x}^{m+n+1}ab}{m+n+1}} \right ) {x}^{m+2\,n+2}y{a}^{2}b+4\,amy{x}^{n+1} \WhittakerM \left ( 1/2\,{\frac {m+2\,n+2}{m+n+1}},1/2\,{\frac {2\,m+3\,n+3}{m+n+1}},-2\,{\frac {{x}^{m+n+1}ab}{m+n+1}} \right ) +ay{m}^{2}{x}^{n+1} \WhittakerM \left ( 1/2\,{\frac {m+2\,n+2}{m+n+1}},1/2\,{\frac {2\,m+3\,n+3}{m+n+1}},-2\,{\frac {{x}^{m+n+1}ab}{m+n+1}} \right ) +ay{m}^{2}{x}^{n+1} \WhittakerM \left ( -1/2\,{\frac {m}{m+n+1}},1/2\,{\frac {2\,m+3\,n+3}{m+n+1}},-2\,{\frac {{x}^{m+n+1}ab}{m+n+1}} \right ) +3\,amy{x}^{n+1} \WhittakerM \left ( -1/2\,{\frac {m}{m+n+1}},1/2\,{\frac {2\,m+3\,n+3}{m+n+1}},-2\,{\frac {{x}^{m+n+1}ab}{m+n+1}} \right ) +2\, \WhittakerM \left ( -1/2\,{\frac {m}{m+n+1}},1/2\,{\frac {2\,m+3\,n+3}{m+n+1}},-2\,{\frac {{x}^{m+n+1}ab}{m+n+1}} \right ) {x}^{2\,n+2\,m+2}{a}^{2}{b}^{2}m+2\, \WhittakerM \left ( -1/2\,{\frac {m}{m+n+1}},1/2\,{\frac {2\,m+3\,n+3}{m+n+1}},-2\,{\frac {{x}^{m+n+1}ab}{m+n+1}} \right ) {x}^{2\,n+2\,m+2}{a}^{2}{b}^{2}n-3\,ab{m}^{2}{x}^{m+n+1} \WhittakerM \left ( 1/2\,{\frac {m+2\,n+2}{m+n+1}},1/2\,{\frac {2\,m+3\,n+3}{m+n+1}},-2\,{\frac {{x}^{m+n+1}ab}{m+n+1}} \right ) -10\,abm{x}^{m+n+1} \WhittakerM \left ( 1/2\,{\frac {m+2\,n+2}{m+n+1}},1/2\,{\frac {2\,m+3\,n+3}{m+n+1}},-2\,{\frac {{x}^{m+n+1}ab}{m+n+1}} \right ) -3\,abm{x}^{m+n+1} \WhittakerM \left ( -1/2\,{\frac {m}{m+n+1}},1/2\,{\frac {2\,m+3\,n+3}{m+n+1}},-2\,{\frac {{x}^{m+n+1}ab}{m+n+1}} \right ) -8\,ab{n}^{2}{x}^{m+n+1} \WhittakerM \left ( 1/2\,{\frac {m+2\,n+2}{m+n+1}},1/2\,{\frac {2\,m+3\,n+3}{m+n+1}},-2\,{\frac {{x}^{m+n+1}ab}{m+n+1}} \right ) -2\,ab{n}^{2}{x}^{m+n+1} \WhittakerM \left ( -1/2\,{\frac {m}{m+n+1}},1/2\,{\frac {2\,m+3\,n+3}{m+n+1}},-2\,{\frac {{x}^{m+n+1}ab}{m+n+1}} \right ) -16\,abn{x}^{m+n+1} \WhittakerM \left ( 1/2\,{\frac {m+2\,n+2}{m+n+1}},1/2\,{\frac {2\,m+3\,n+3}{m+n+1}},-2\,{\frac {{x}^{m+n+1}ab}{m+n+1}} \right ) -4\,abn{x}^{m+n+1} \WhittakerM \left ( -1/2\,{\frac {m}{m+n+1}},1/2\,{\frac {2\,m+3\,n+3}{m+n+1}},-2\,{\frac {{x}^{m+n+1}ab}{m+n+1}} \right ) -ab{m}^{2}{x}^{m+n+1} \WhittakerM \left ( -1/2\,{\frac {m}{m+n+1}},1/2\,{\frac {2\,m+3\,n+3}{m+n+1}},-2\,{\frac {{x}^{m+n+1}ab}{m+n+1}} \right ) -5\, \WhittakerM \left ( -1/2\,{\frac {m}{m+n+1}},1/2\,{\frac {2\,m+3\,n+3}{m+n+1}},-2\,{\frac {{x}^{m+n+1}ab}{m+n+1}} \right ) m+36\,n \left ( -2\,{\frac {{x}^{m+n+1}ab}{m+n+1}} \right ) ^{1/2\,{\frac {3\,m+4\,n+4}{m+n+1}}}{{\rm e}^{{\frac {{x}^{m+n+1}ab}{m+n+1}}}}+11\,{m}^{2} \left ( -2\,{\frac {{x}^{m+n+1}ab}{m+n+1}} \right ) ^{1/2\,{\frac {3\,m+4\,n+4}{m+n+1}}}{{\rm e}^{{\frac {{x}^{m+n+1}ab}{m+n+1}}}}+4\,ay{n}^{2}{x}^{n+1} \WhittakerM \left ( 1/2\,{\frac {m+2\,n+2}{m+n+1}},1/2\,{\frac {2\,m+3\,n+3}{m+n+1}},-2\,{\frac {{x}^{m+n+1}ab}{m+n+1}} \right ) +4\,ay{x}^{n+1} \WhittakerM \left ( 1/2\,{\frac {m+2\,n+2}{m+n+1}},1/2\,{\frac {2\,m+3\,n+3}{m+n+1}},-2\,{\frac {{x}^{m+n+1}ab}{m+n+1}} \right ) +2\,ay{x}^{n+1} \WhittakerM \left ( -1/2\,{\frac {m}{m+n+1}},1/2\,{\frac {2\,m+3\,n+3}{m+n+1}},-2\,{\frac {{x}^{m+n+1}ab}{m+n+1}} \right ) +2\, \WhittakerM \left ( -1/2\,{\frac {m}{m+n+1}},1/2\,{\frac {2\,m+3\,n+3}{m+n+1}},-2\,{\frac {{x}^{m+n+1}ab}{m+n+1}} \right ) {x}^{2\,n+2\,m+2}{a}^{2}{b}^{2}-8\,ab{x}^{m+n+1} \WhittakerM \left ( 1/2\,{\frac {m+2\,n+2}{m+n+1}},1/2\,{\frac {2\,m+3\,n+3}{m+n+1}},-2\,{\frac {{x}^{m+n+1}ab}{m+n+1}} \right ) -2\,ab{x}^{m+n+1} \WhittakerM \left ( -1/2\,{\frac {m}{m+n+1}},1/2\,{\frac {2\,m+3\,n+3}{m+n+1}},-2\,{\frac {{x}^{m+n+1}ab}{m+n+1}} \right ) -6\, \WhittakerM \left ( -1/2\,{\frac {m}{m+n+1}},1/2\,{\frac {2\,m+3\,n+3}{m+n+1}},-2\,{\frac {{x}^{m+n+1}ab}{m+n+1}} \right ) n-{m}^{3} \WhittakerM \left ( 1/2\,{\frac {m+2\,n+2}{m+n+1}},1/2\,{\frac {2\,m+3\,n+3}{m+n+1}},-2\,{\frac {{x}^{m+n+1}ab}{m+n+1}} \right ) -{m}^{3} \WhittakerM \left ( -1/2\,{\frac {m}{m+n+1}},1/2\,{\frac {2\,m+3\,n+3}{m+n+1}},-2\,{\frac {{x}^{m+n+1}ab}{m+n+1}} \right ) -12\,n \WhittakerM \left ( 1/2\,{\frac {m+2\,n+2}{m+n+1}},1/2\,{\frac {2\,m+3\,n+3}{m+n+1}},-2\,{\frac {{x}^{m+n+1}ab}{m+n+1}} \right ) -5\,{m}^{2} \WhittakerM \left ( 1/2\,{\frac {m+2\,n+2}{m+n+1}},1/2\,{\frac {2\,m+3\,n+3}{m+n+1}},-2\,{\frac {{x}^{m+n+1}ab}{m+n+1}} \right ) -4\,{m}^{2} \WhittakerM \left ( -1/2\,{\frac {m}{m+n+1}},1/2\,{\frac {2\,m+3\,n+3}{m+n+1}},-2\,{\frac {{x}^{m+n+1}ab}{m+n+1}} \right ) -12\,{n}^{2} \WhittakerM \left ( 1/2\,{\frac {m+2\,n+2}{m+n+1}},1/2\,{\frac {2\,m+3\,n+3}{m+n+1}},-2\,{\frac {{x}^{m+n+1}ab}{m+n+1}} \right ) -6\,{n}^{2} \WhittakerM \left ( -1/2\,{\frac {m}{m+n+1}},1/2\,{\frac {2\,m+3\,n+3}{m+n+1}},-2\,{\frac {{x}^{m+n+1}ab}{m+n+1}} \right ) -8\,m \WhittakerM \left ( 1/2\,{\frac {m+2\,n+2}{m+n+1}},1/2\,{\frac {2\,m+3\,n+3}{m+n+1}},-2\,{\frac {{x}^{m+n+1}ab}{m+n+1}} \right ) -4\,{n}^{3} \WhittakerM \left ( 1/2\,{\frac {m+2\,n+2}{m+n+1}},1/2\,{\frac {2\,m+3\,n+3}{m+n+1}},-2\,{\frac {{x}^{m+n+1}ab}{m+n+1}} \right ) -2\,{n}^{3} \WhittakerM \left ( -1/2\,{\frac {m}{m+n+1}},1/2\,{\frac {2\,m+3\,n+3}{m+n+1}},-2\,{\frac {{x}^{m+n+1}ab}{m+n+1}} \right ) +20\,m{n}^{2} \left ( -2\,{\frac {{x}^{m+n+1}ab}{m+n+1}} \right ) ^{1/2\,{\frac {3\,m+4\,n+4}{m+n+1}}}{{\rm e}^{{\frac {{x}^{m+n+1}ab}{m+n+1}}}}+40\,mn \left ( -2\,{\frac {{x}^{m+n+1}ab}{m+n+1}} \right ) ^{1/2\,{\frac {3\,m+4\,n+4}{m+n+1}}}{{\rm e}^{{\frac {{x}^{m+n+1}ab}{m+n+1}}}}+11\,{m}^{2}n \left ( -2\,{\frac {{x}^{m+n+1}ab}{m+n+1}} \right ) ^{1/2\,{\frac {3\,m+4\,n+4}{m+n+1}}}{{\rm e}^{{\frac {{x}^{m+n+1}ab}{m+n+1}}}}+12\,{n}^{3} \left ( -2\,{\frac {{x}^{m+n+1}ab}{m+n+1}} \right ) ^{1/2\,{\frac {3\,m+4\,n+4}{m+n+1}}}{{\rm e}^{{\frac {{x}^{m+n+1}ab}{m+n+1}}}}+2\,{m}^{3} \left ( -2\,{\frac {{x}^{m+n+1}ab}{m+n+1}} \right ) ^{1/2\,{\frac {3\,m+4\,n+4}{m+n+1}}}{{\rm e}^{{\frac {{x}^{m+n+1}ab}{m+n+1}}}}-5\,m{n}^{2} \WhittakerM \left ( -1/2\,{\frac {m}{m+n+1}},1/2\,{\frac {2\,m+3\,n+3}{m+n+1}},-2\,{\frac {{x}^{m+n+1}ab}{m+n+1}} \right ) -4\,{m}^{2} \WhittakerM \left ( -1/2\,{\frac {m}{m+n+1}},1/2\,{\frac {2\,m+3\,n+3}{m+n+1}},-2\,{\frac {{x}^{m+n+1}ab}{m+n+1}} \right ) n-5\,{m}^{2}n \WhittakerM \left ( 1/2\,{\frac {m+2\,n+2}{m+n+1}},1/2\,{\frac {2\,m+3\,n+3}{m+n+1}},-2\,{\frac {{x}^{m+n+1}ab}{m+n+1}} \right ) -8\,m{n}^{2} \WhittakerM \left ( 1/2\,{\frac {m+2\,n+2}{m+n+1}},1/2\,{\frac {2\,m+3\,n+3}{m+n+1}},-2\,{\frac {{x}^{m+n+1}ab}{m+n+1}} \right ) -16\,mn \WhittakerM \left ( 1/2\,{\frac {m+2\,n+2}{m+n+1}},1/2\,{\frac {2\,m+3\,n+3}{m+n+1}},-2\,{\frac {{x}^{m+n+1}ab}{m+n+1}} \right ) -10\,m \WhittakerM \left ( -1/2\,{\frac {m}{m+n+1}},1/2\,{\frac {2\,m+3\,n+3}{m+n+1}},-2\,{\frac {{x}^{m+n+1}ab}{m+n+1}} \right ) n+36\,{n}^{2} \left ( -2\,{\frac {{x}^{m+n+1}ab}{m+n+1}} \right ) ^{1/2\,{\frac {3\,m+4\,n+4}{m+n+1}}}{{\rm e}^{{\frac {{x}^{m+n+1}ab}{m+n+1}}}}+20\,m \left ( -2\,{\frac {{x}^{m+n+1}ab}{m+n+1}} \right ) ^{1/2\,{\frac {3\,m+4\,n+4}{m+n+1}}}{{\rm e}^{{\frac {{x}^{m+n+1}ab}{m+n+1}}}}+12\, \left ( -2\,{\frac {{x}^{m+n+1}ab}{m+n+1}} \right ) ^{1/2\,{\frac {3\,m+4\,n+4}{m+n+1}}}{{\rm e}^{{\frac {{x}^{m+n+1}ab}{m+n+1}}}}-4\, \WhittakerM \left ( 1/2\,{\frac {m+2\,n+2}{m+n+1}},1/2\,{\frac {2\,m+3\,n+3}{m+n+1}},-2\,{\frac {{x}^{m+n+1}ab}{m+n+1}} \right ) -2\, \WhittakerM \left ( -1/2\,{\frac {m}{m+n+1}},1/2\,{\frac {2\,m+3\,n+3}{m+n+1}},-2\,{\frac {{x}^{m+n+1}ab}{m+n+1}} \right ) -10\,abmn{x}^{m+n+1} \WhittakerM \left ( 1/2\,{\frac {m+2\,n+2}{m+n+1}},1/2\,{\frac {2\,m+3\,n+3}{m+n+1}},-2\,{\frac {{x}^{m+n+1}ab}{m+n+1}} \right ) -3\,abmn{x}^{m+n+1} \WhittakerM \left ( -1/2\,{\frac {m}{m+n+1}},1/2\,{\frac {2\,m+3\,n+3}{m+n+1}},-2\,{\frac {{x}^{m+n+1}ab}{m+n+1}} \right ) -2\, \WhittakerM \left ( -1/2\,{\frac {m}{m+n+1}},1/2\,{\frac {2\,m+3\,n+3}{m+n+1}},-2\,{\frac {{x}^{m+n+1}ab}{m+n+1}} \right ) {x}^{m+2\,n+2}y{a}^{2}bn+4\,amyn{x}^{n+1} \WhittakerM \left ( 1/2\,{\frac {m+2\,n+2}{m+n+1}},1/2\,{\frac {2\,m+3\,n+3}{m+n+1}},-2\,{\frac {{x}^{m+n+1}ab}{m+n+1}} \right ) +3\,amyn{x}^{n+1} \WhittakerM \left ( -1/2\,{\frac {m}{m+n+1}},1/2\,{\frac {2\,m+3\,n+3}{m+n+1}},-2\,{\frac {{x}^{m+n+1}ab}{m+n+1}} \right ) -2\, \WhittakerM \left ( -1/2\,{\frac {m}{m+n+1}},1/2\,{\frac {2\,m+3\,n+3}{m+n+1}},-2\,{\frac {{x}^{m+n+1}ab}{m+n+1}} \right ) {x}^{m+2\,n+2}y{a}^{2}bm \right ) ^{-1}} \right ) \]
____________________________________________________________________________________
Added January 2, 2019.
Problem 2.2.5.9 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + \left ( (n+1)x^n y^2 - a x^{n+m+1} y + a x^m \right ) w_y = 0 \]
Mathematica ✗
ClearAll[w, x, y, n, a, b, m]; pde = D[w[x, y], x] + ((n + 1)*x^n*y^2 - a*x^(n + m + 1)*y + a*x^m)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \text {Failed} \]
Maple ✓
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m'; pde := diff(w(x,y),x)+ ((n+1)*x^n*y^2 - a*x^(n+m+1)* y + a*x^m)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{1 \left ( -2\,{\mbox {$_1$F$_1$}({\frac {-n+m}{m+n+2}};\,{\frac {m+1}{m+n+2}};\,{\frac {{x}^{2}{x}^{n}{x}^{m}a}{m+n+2}})}{x}^{m}amnx+{\mbox {$_1$F$_1$}(2\,{\frac {m+1}{m+n+2}};\,{\frac {2\,m+3+n}{m+n+2}};\,{\frac {{x}^{2}{x}^{n}{x}^{m}a}{m+n+2}})}{x}^{m}amnx-3\,{\mbox {$_1$F$_1$}({\frac {-n+m}{m+n+2}};\,{\frac {m+1}{m+n+2}};\,{\frac {{x}^{2}{x}^{n}{x}^{m}a}{m+n+2}})}{x}^{m}ax-{\mbox {$_1$F$_1$}({\frac {-n+m}{m+n+2}};\,{\frac {m+1}{m+n+2}};\,{\frac {{x}^{2}{x}^{n}{x}^{m}a}{m+n+2}})}{x}^{m}a{m}^{2}x+{\mbox {$_1$F$_1$}(2\,{\frac {m+1}{m+n+2}};\,{\frac {2\,m+3+n}{m+n+2}};\,{\frac {{x}^{2}{x}^{n}{x}^{m}a}{m+n+2}})}{x}^{m}a{m}^{2}x-2\,{\mbox {$_1$F$_1$}(2\,{\frac {m+1}{m+n+2}};\,{\frac {2\,m+3+n}{m+n+2}};\,{\frac {{x}^{2}{x}^{n}{x}^{m}a}{m+n+2}})}{x}^{m}a{n}^{2}x+{\mbox {$_1$F$_1$}({\frac {-n+m}{m+n+2}};\,{\frac {m+1}{m+n+2}};\,{\frac {{x}^{2}{x}^{n}{x}^{m}a}{m+n+2}})}y{m}^{2}n+2\,{\mbox {$_1$F$_1$}({\frac {-n+m}{m+n+2}};\,{\frac {m+1}{m+n+2}};\,{\frac {{x}^{2}{x}^{n}{x}^{m}a}{m+n+2}})}ym{n}^{2}+6\,{\mbox {$_1$F$_1$}({\frac {-n+m}{m+n+2}};\,{\frac {m+1}{m+n+2}};\,{\frac {{x}^{2}{x}^{n}{x}^{m}a}{m+n+2}})}ymn-4\,{\mbox {$_1$F$_1$}({\frac {-n+m}{m+n+2}};\,{\frac {m+1}{m+n+2}};\,{\frac {{x}^{2}{x}^{n}{x}^{m}a}{m+n+2}})}{x}^{m}amx-2\,{\mbox {$_1$F$_1$}({\frac {-n+m}{m+n+2}};\,{\frac {m+1}{m+n+2}};\,{\frac {{x}^{2}{x}^{n}{x}^{m}a}{m+n+2}})}{x}^{m}anx+3\,{\mbox {$_1$F$_1$}(2\,{\frac {m+1}{m+n+2}};\,{\frac {2\,m+3+n}{m+n+2}};\,{\frac {{x}^{2}{x}^{n}{x}^{m}a}{m+n+2}})}{x}^{m}amx-3\,{\mbox {$_1$F$_1$}(2\,{\frac {m+1}{m+n+2}};\,{\frac {2\,m+3+n}{m+n+2}};\,{\frac {{x}^{2}{x}^{n}{x}^{m}a}{m+n+2}})}{x}^{m}anx+{\mbox {$_1$F$_1$}({\frac {-n+m}{m+n+2}};\,{\frac {m+1}{m+n+2}};\,{\frac {{x}^{2}{x}^{n}{x}^{m}a}{m+n+2}})}y{m}^{2}+2\,{\mbox {$_1$F$_1$}({\frac {-n+m}{m+n+2}};\,{\frac {m+1}{m+n+2}};\,{\frac {{x}^{2}{x}^{n}{x}^{m}a}{m+n+2}})}y{n}^{2}+4\,{\mbox {$_1$F$_1$}({\frac {-n+m}{m+n+2}};\,{\frac {m+1}{m+n+2}};\,{\frac {{x}^{2}{x}^{n}{x}^{m}a}{m+n+2}})}ym+5\,{\mbox {$_1$F$_1$}({\frac {-n+m}{m+n+2}};\,{\frac {m+1}{m+n+2}};\,{\frac {{x}^{2}{x}^{n}{x}^{m}a}{m+n+2}})}yn+3\,{\mbox {$_1$F$_1$}({\frac {-n+m}{m+n+2}};\,{\frac {m+1}{m+n+2}};\,{\frac {{x}^{2}{x}^{n}{x}^{m}a}{m+n+2}})}y \right ) \left ( -3\,{\mbox {$_1$F$_1$}({\frac {m+1}{m+n+2}};\,{\frac {m+2\,n+3}{m+n+2}};\,{\frac {{x}^{2}{x}^{n}{x}^{m}a}{m+n+2}})}{x}^{n}{x}^{m}a{x}^{2}+{x}^{m}{x}^{n}{x}^{2}a{\mbox {$_1$F$_1$}({\frac {2\,m+3+n}{m+n+2}};\,{\frac {2\,m+3\,n+5}{m+n+2}};\,{\frac {{x}^{2}{x}^{n}{x}^{m}a}{m+n+2}})}+{\mbox {$_1$F$_1$}({\frac {m+1}{m+n+2}};\,{\frac {m+2\,n+3}{m+n+2}};\,{\frac {{x}^{2}{x}^{n}{x}^{m}a}{m+n+2}})}{x}^{n}y{m}^{2}x+2\,{\mbox {$_1$F$_1$}({\frac {m+1}{m+n+2}};\,{\frac {m+2\,n+3}{m+n+2}};\,{\frac {{x}^{2}{x}^{n}{x}^{m}a}{m+n+2}})}{x}^{n}y{n}^{2}x+4\,{\mbox {$_1$F$_1$}({\frac {m+1}{m+n+2}};\,{\frac {m+2\,n+3}{m+n+2}};\,{\frac {{x}^{2}{x}^{n}{x}^{m}a}{m+n+2}})}{x}^{n}ymx+5\,{\mbox {$_1$F$_1$}({\frac {m+1}{m+n+2}};\,{\frac {m+2\,n+3}{m+n+2}};\,{\frac {{x}^{2}{x}^{n}{x}^{m}a}{m+n+2}})}{x}^{n}ynx+3\,{\mbox {$_1$F$_1$}({\frac {m+1}{m+n+2}};\,{\frac {m+2\,n+3}{m+n+2}};\,{\frac {{x}^{2}{x}^{n}{x}^{m}a}{m+n+2}})}+3\,{\mbox {$_1$F$_1$}({\frac {m+1}{m+n+2}};\,{\frac {m+2\,n+3}{m+n+2}};\,{\frac {{x}^{2}{x}^{n}{x}^{m}a}{m+n+2}})}{x}^{n}yx+{\mbox {$_1$F$_1$}({\frac {m+1}{m+n+2}};\,{\frac {m+2\,n+3}{m+n+2}};\,{\frac {{x}^{2}{x}^{n}{x}^{m}a}{m+n+2}})}{m}^{2}n+2\,{\mbox {$_1$F$_1$}({\frac {m+1}{m+n+2}};\,{\frac {m+2\,n+3}{m+n+2}};\,{\frac {{x}^{2}{x}^{n}{x}^{m}a}{m+n+2}})}m{n}^{2}+6\,{\mbox {$_1$F$_1$}({\frac {m+1}{m+n+2}};\,{\frac {m+2\,n+3}{m+n+2}};\,{\frac {{x}^{2}{x}^{n}{x}^{m}a}{m+n+2}})}mn+{\mbox {$_1$F$_1$}({\frac {m+1}{m+n+2}};\,{\frac {m+2\,n+3}{m+n+2}};\,{\frac {{x}^{2}{x}^{n}{x}^{m}a}{m+n+2}})}{m}^{2}+2\,{\mbox {$_1$F$_1$}({\frac {m+1}{m+n+2}};\,{\frac {m+2\,n+3}{m+n+2}};\,{\frac {{x}^{2}{x}^{n}{x}^{m}a}{m+n+2}})}{n}^{2}+4\,{\mbox {$_1$F$_1$}({\frac {m+1}{m+n+2}};\,{\frac {m+2\,n+3}{m+n+2}};\,{\frac {{x}^{2}{x}^{n}{x}^{m}a}{m+n+2}})}m+5\,{\mbox {$_1$F$_1$}({\frac {m+1}{m+n+2}};\,{\frac {m+2\,n+3}{m+n+2}};\,{\frac {{x}^{2}{x}^{n}{x}^{m}a}{m+n+2}})}n+{\mbox {$_1$F$_1$}({\frac {m+1}{m+n+2}};\,{\frac {m+2\,n+3}{m+n+2}};\,{\frac {{x}^{2}{x}^{n}{x}^{m}a}{m+n+2}})}{x}^{n}y{m}^{2}nx+2\,{\mbox {$_1$F$_1$}({\frac {m+1}{m+n+2}};\,{\frac {m+2\,n+3}{m+n+2}};\,{\frac {{x}^{2}{x}^{n}{x}^{m}a}{m+n+2}})}{x}^{n}ym{n}^{2}x+6\,{\mbox {$_1$F$_1$}({\frac {m+1}{m+n+2}};\,{\frac {m+2\,n+3}{m+n+2}};\,{\frac {{x}^{2}{x}^{n}{x}^{m}a}{m+n+2}})}{x}^{n}ymnx-2\,{\mbox {$_1$F$_1$}({\frac {m+1}{m+n+2}};\,{\frac {m+2\,n+3}{m+n+2}};\,{\frac {{x}^{2}{x}^{n}{x}^{m}a}{m+n+2}})}{x}^{n}{x}^{m}amn{x}^{2}-{\mbox {$_1$F$_1$}({\frac {m+1}{m+n+2}};\,{\frac {m+2\,n+3}{m+n+2}};\,{\frac {{x}^{2}{x}^{n}{x}^{m}a}{m+n+2}})}{x}^{n}{x}^{m}a{m}^{2}{x}^{2}+{\mbox {$_1$F$_1$}({\frac {2\,m+3+n}{m+n+2}};\,{\frac {2\,m+3\,n+5}{m+n+2}};\,{\frac {{x}^{2}{x}^{n}{x}^{m}a}{m+n+2}})}{x}^{n}{x}^{m}a{m}^{2}{x}^{2}-4\,{\mbox {$_1$F$_1$}({\frac {m+1}{m+n+2}};\,{\frac {m+2\,n+3}{m+n+2}};\,{\frac {{x}^{2}{x}^{n}{x}^{m}a}{m+n+2}})}{x}^{n}{x}^{m}am{x}^{2}-2\,{\mbox {$_1$F$_1$}({\frac {m+1}{m+n+2}};\,{\frac {m+2\,n+3}{m+n+2}};\,{\frac {{x}^{2}{x}^{n}{x}^{m}a}{m+n+2}})}{x}^{n}{x}^{m}an{x}^{2}+2\,{\mbox {$_1$F$_1$}({\frac {2\,m+3+n}{m+n+2}};\,{\frac {2\,m+3\,n+5}{m+n+2}};\,{\frac {{x}^{2}{x}^{n}{x}^{m}a}{m+n+2}})}{x}^{n}{x}^{m}am{x}^{2} \right ) ^{-1}} \right ) \]
____________________________________________________________________________________
Added January 2, 2019.
Problem 2.2.5.10 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + \left ( a x^n y^2 + b x^m y+ b c x^m -a c^2 x^n \right ) w_y = 0 \]
Mathematica ✗
ClearAll[w, x, y, n, a, b, m, c]; pde = D[w[x, y], x] + (a*x^n*y^2 + b*x^m*y + b*c*x^m - a*c^2*x^n)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \text {\$Aborted} \] Timed out
Maple ✓
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c'; pde := diff(w(x,y),x)+ (a*x^n*y^2 + b*x^m*y+ b*c*x^m -a*c^2*x^n)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac {1}{c+y} \left ( \int \!{{\rm e}^{{\frac {x \left ( -2\,ca{x}^{n}m+{x}^{m}bn-2\,{x}^{n}ac+{x}^{m}b \right ) }{ \left ( m+1 \right ) \left ( n+1 \right ) }}}}{x}^{n}a\,{\rm d}xy+\int \!{{\rm e}^{{\frac {x \left ( -2\,ca{x}^{n}m+{x}^{m}bn-2\,{x}^{n}ac+{x}^{m}b \right ) }{ \left ( m+1 \right ) \left ( n+1 \right ) }}}}{x}^{n}a\,{\rm d}xc+{{\rm e}^{-{\frac {x \left ( 2\,ca{x}^{n}m+2\,{x}^{n}ac-{x}^{m}bn-{x}^{m}b \right ) }{ \left ( m+1 \right ) \left ( n+1 \right ) }}}} \right ) } \right ) \]
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Added January 2, 2019.
Problem 2.2.5.11 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + \left ( a x^n y^2-a x^n (b x^m +c) y+ b m x^{m-1} \right ) w_y = 0 \]
Mathematica ✗
ClearAll[w, x, y, n, a, b, m, c]; pde = D[w[x, y], x] + (a*x^n*y^2 - a*x^n*(b*x^m + c)*y + b*m*x^(m - 1))*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \text {Failed} \]
Maple ✗
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c'; pde := diff(w(x,y),x)+ (a*x^n*y^2-a*x^n*(b*x^m +c)*y+ b*m*x^(m-1))*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ \text { sol=() } \]
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Added January 2, 2019.
Problem 2.2.5.12 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x - \left (a n x^{n-1} y^2 - c x^m (a x^n+b) + c x^m \right ) w_y = 0 \]
Mathematica ✗
ClearAll[w, x, y, n, a, b, m, c]; pde = D[w[x, y], x] - (a*n*x^(n - 1)*y^2 - c*x^m*(a*x^n + b) + c*x^m)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \text {Failed} \]
Maple ✗
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c'; pde := diff(w(x,y),x)- (a*n*x^(n-1)*y^2 - c*x^m*(a*x^n+b) + c*x^m)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ \text { sol=() } \]
____________________________________________________________________________________
Added January 2, 2019.
Problem 2.2.5.13 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + \left ( a x^n y^2+b x^m y+ c k x^{k-1}-b c x^{m+k}-a c^2 x^{n+2 k} \right ) w_y = 0 \]
Mathematica ✗
ClearAll[w, x, y, n, a, b, m, c, k]; pde = D[w[x, y], x] + (a*x^n*y^2 + b*x^m*y + c*k*x^(k - 1) - b*c*x^(m + k) - a*c^2*x^(n + 2*k))*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \text {Failed} \]
Maple ✗
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k'; pde := diff(w(x,y),x)+ (a*x^n*y^2+b*x^m*y+ c*k*x^(k-1)-b*c*x^(m+k)-a*c^2*x^(n+2*k))*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ \text { sol=() } \]
____________________________________________________________________________________
Added January 2, 2019.
Problem 2.2.5.14 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + \left ( a x^{2 n+1} y^3 + b x^{-n-2} \right ) w_y = 0 \]
Mathematica ✗
ClearAll[w, x, y, n, a, b, m, c, k]; pde = D[w[x, y], x] + (a*x^(2*n + 1)*y^3 + b*x^(-n - 2))*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \text {Failed} \]
Maple ✓
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k'; pde := diff(w(x,y),x)+ (a*x^(2*n+1)*y^3 + b*x^(-n-2))*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( \ln \left ( x \right ) -\sum _{{\it \_R}=\RootOf \left ( a{{\it \_Z}}^{3}+{\it \_Z}\, \left ( n+1 \right ) +b \right ) }{\frac {\ln \left ( y{x}^{n}x-{\it \_R} \right ) }{3\,{{\it \_R}}^{2}a+n+1}} \right ) \] Solution contains RootOf
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Added January 2, 2019.
Problem 2.2.5.15 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + \left ( a x^n y^3 + 3 a b x^{n+m} y^2 - b m x^{m-1} - 2 a b^3 x^{n+3 m} \right ) w_y = 0 \]
Mathematica ✓
ClearAll[w, x, y, n, a, b, m, c, k]; pde = D[w[x, y], x] + (a*x^n*y^3 + 3*a*b*x^(n + m)*y^2 - b*m*x^(m - 1) - 2*a*b^3*x^(n + 3*m))*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to c_1\left (\frac {6^{-\frac {n}{2 m+n+1}-\frac {1}{2 m+n+1}} (2 m+n+1)^{-\frac {2 m}{2 m+n+1}} b^{-\frac {2 n}{2 m+n+1}-\frac {2}{2 m+n+1}} e^{-\frac {6 a b^2 x^{2 m+n+1}}{2 m+n+1}} \left (-2 y^2 a^{\frac {2 m}{2 m+n+1}} e^{\frac {6 a b^2 x^{2 m+n+1}}{2 m+n+1}} \text {Gamma}\left (\frac {n+1}{2 m+n+1},\frac {6 a b^2 x^{2 m+n+1}}{2 m+n+1}\right )-2 b^2 x^{2 m} a^{\frac {2 m}{2 m+n+1}} e^{\frac {6 a b^2 x^{2 m+n+1}}{2 m+n+1}} \text {Gamma}\left (\frac {n+1}{2 m+n+1},\frac {6 a b^2 x^{2 m+n+1}}{2 m+n+1}\right )-4 b y x^m a^{\frac {2 m}{2 m+n+1}} e^{\frac {6 a b^2 x^{2 m+n+1}}{2 m+n+1}} \text {Gamma}\left (\frac {n+1}{2 m+n+1},\frac {6 a b^2 x^{2 m+n+1}}{2 m+n+1}\right )+6^{\frac {n}{2 m+n+1}+\frac {1}{2 m+n+1}} (2 m+n+1)^{\frac {2 m}{2 m+n+1}} b^{\frac {2 n}{2 m+n+1}+\frac {2}{2 m+n+1}}\right )}{x^{2 m} b^{\frac {4 m}{2 m+n+1}+\frac {2 n}{2 m+n+1}+\frac {2}{2 m+n+1}}+2 b y x^m+y^2}\right )\right \}\right \} \]
Maple ✓
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k'; pde := diff(w(x,y),x)+ (a*x^n*y^3 + 3*a*b*x^(n+m)*y^2 - b*m*x^(m-1) - 2*a*b^3*x^(n+3*m))*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1}{{b}^{2} \left ( 3\,{y}^{2}+14\,bm{x}^{m}y+3\,{y}^{2}{n}^{3}+4\,{y}^{2}{m}^{2}+9\,{y}^{2}{n}^{2}+7\,{y}^{2}m+9\,{y}^{2}n+6\,y{x}^{m}b{n}^{3}+18\,y{x}^{m}b{n}^{2}+18\,y{x}^{m}bn+8\,b{m}^{2}{x}^{m}y+6\,b{x}^{m}y+3\,{x}^{2\,m}{b}^{2}+4\,{x}^{2\,m}{b}^{2}{m}^{2}n+7\,{x}^{2\,m}{b}^{2}m{n}^{2}+14\,{x}^{2\,m}{b}^{2}mn+8\,y{x}^{m}b{m}^{2}n+14\,y{x}^{m}bm{n}^{2}+28\,y{x}^{m}bmn+4\,{x}^{2\,m}{b}^{2}{m}^{2}+9\,{x}^{2\,m}{b}^{2}{n}^{2}+7\,{x}^{2\,m}{b}^{2}m+9\,{x}^{2\,m}{b}^{2}n+3\,{x}^{2\,m}{b}^{2}{n}^{3}+4\,{y}^{2}{m}^{2}n+7\,{y}^{2}m{n}^{2}+14\,{y}^{2}mn \right ) } \left ( 12\,{{\rm e}^{-3\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}}}}{27}^{-{\frac {m}{2\,m+n+1}}}{9}^{-{\frac {n+1}{2\,m+n+1}}}{2}^{-{\frac {m+n+1}{2\,m+n+1}}} \left ( {\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) ^{-{\frac {m+n+1}{2\,m+n+1}}}{x}^{n+1} \WhittakerM \left ( -{\frac {m}{2\,m+n+1}},1/2\,{\frac {4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) a{b}^{2}mn{y}^{2}+24\,{{\rm e}^{-3\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}}}}{27}^{-{\frac {m}{2\,m+n+1}}}{9}^{-{\frac {n+1}{2\,m+n+1}}}{2}^{-{\frac {m+n+1}{2\,m+n+1}}} \left ( {\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) ^{-{\frac {m+n+1}{2\,m+n+1}}}{x}^{m+n+1} \WhittakerM \left ( -{\frac {m}{2\,m+n+1}},1/2\,{\frac {4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) a{b}^{3}mny+42\,{{\rm e}^{-6\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}}}}{9}^{-{\frac {m}{2\,m+n+1}}}{3}^{-{\frac {n+1}{2\,m+n+1}}}{b}^{2}mn+12\,{{\rm e}^{-6\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}}}}{9}^{-{\frac {m}{2\,m+n+1}}}{3}^{-{\frac {n+1}{2\,m+n+1}}}{b}^{2}{m}^{2}n+21\,{{\rm e}^{-6\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}}}}{9}^{-{\frac {m}{2\,m+n+1}}}{3}^{-{\frac {n+1}{2\,m+n+1}}}{b}^{2}m{n}^{2}+6\,{{\rm e}^{-3\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}}}}{27}^{-{\frac {m}{2\,m+n+1}}}{9}^{-{\frac {n+1}{2\,m+n+1}}}{2}^{-{\frac {m+n+1}{2\,m+n+1}}} \left ( {\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) ^{-{\frac {m+n+1}{2\,m+n+1}}}{x}^{n+1} \WhittakerM \left ( -{\frac {m}{2\,m+n+1}},1/2\,{\frac {4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) {y}^{2}a{b}^{2}n+12\,{{\rm e}^{-3\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}}}}{27}^{-{\frac {m}{2\,m+n+1}}}{9}^{-{\frac {n+1}{2\,m+n+1}}}{2}^{-{\frac {m+n+1}{2\,m+n+1}}} \left ( {\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) ^{-{\frac {m+n+1}{2\,m+n+1}}}{x}^{2\,m+n+1} \WhittakerM \left ( -{\frac {m}{2\,m+n+1}},1/2\,{\frac {4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) a{b}^{4}{m}^{2}+5\,{{\rm e}^{-3\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}}}}{27}^{-{\frac {m}{2\,m+n+1}}}{9}^{-{\frac {n+1}{2\,m+n+1}}}{2}^{-{\frac {m+n+1}{2\,m+n+1}}} \left ( {\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) ^{-{\frac {m+n+1}{2\,m+n+1}}}{x}^{-2\,m} \WhittakerM \left ( -{\frac {m}{2\,m+n+1}},1/2\,{\frac {4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) {y}^{2}m{n}^{2}+16\,{{\rm e}^{-3\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}}}}{27}^{-{\frac {m}{2\,m+n+1}}}{9}^{-{\frac {n+1}{2\,m+n+1}}}{2}^{-{\frac {m+n+1}{2\,m+n+1}}} \left ( {\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) ^{-{\frac {m+n+1}{2\,m+n+1}}}{x}^{-m} \WhittakerM \left ( {\frac {m+n+1}{2\,m+n+1}},1/2\,{\frac {4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) ybm+10\,{{\rm e}^{-3\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}}}}{27}^{-{\frac {m}{2\,m+n+1}}}{9}^{-{\frac {n+1}{2\,m+n+1}}}{2}^{-{\frac {m+n+1}{2\,m+n+1}}} \left ( {\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) ^{-{\frac {m+n+1}{2\,m+n+1}}}{x}^{-2\,m} \WhittakerM \left ( {\frac {m+n+1}{2\,m+n+1}},1/2\,{\frac {4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) {y}^{2}{m}^{2}n+8\,{{\rm e}^{-3\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}}}}{27}^{-{\frac {m}{2\,m+n+1}}}{9}^{-{\frac {n+1}{2\,m+n+1}}}{2}^{-{\frac {m+n+1}{2\,m+n+1}}} \left ( {\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) ^{-{\frac {m+n+1}{2\,m+n+1}}}{x}^{-2\,m} \WhittakerM \left ( {\frac {m+n+1}{2\,m+n+1}},1/2\,{\frac {4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) {y}^{2}m{n}^{2}+10\,{{\rm e}^{-3\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}}}}{27}^{-{\frac {m}{2\,m+n+1}}}{9}^{-{\frac {n+1}{2\,m+n+1}}}{2}^{-{\frac {m+n+1}{2\,m+n+1}}} \left ( {\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) ^{-{\frac {m+n+1}{2\,m+n+1}}}{x}^{-2\,m} \WhittakerM \left ( -{\frac {m}{2\,m+n+1}},1/2\,{\frac {4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) {y}^{2}mn+8\,{{\rm e}^{-3\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}}}}{27}^{-{\frac {m}{2\,m+n+1}}}{9}^{-{\frac {n+1}{2\,m+n+1}}}{2}^{-{\frac {m+n+1}{2\,m+n+1}}} \left ( {\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) ^{-{\frac {m+n+1}{2\,m+n+1}}}{x}^{-m} \WhittakerM \left ( -{\frac {m}{2\,m+n+1}},1/2\,{\frac {4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) yb{m}^{3}+3\,{{\rm e}^{-3\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}}}}{27}^{-{\frac {m}{2\,m+n+1}}}{9}^{-{\frac {n+1}{2\,m+n+1}}}{2}^{-{\frac {m+n+1}{2\,m+n+1}}} \left ( {\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) ^{-{\frac {m+n+1}{2\,m+n+1}}}{x}^{n+1} \WhittakerM \left ( -{\frac {m}{2\,m+n+1}},1/2\,{\frac {4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) {y}^{2}a{b}^{2}+8\,{{\rm e}^{-3\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}}}}{27}^{-{\frac {m}{2\,m+n+1}}}{9}^{-{\frac {n+1}{2\,m+n+1}}}{2}^{-{\frac {m+n+1}{2\,m+n+1}}} \left ( {\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) ^{-{\frac {m+n+1}{2\,m+n+1}}}{x}^{-2\,m} \WhittakerM \left ( -{\frac {m}{2\,m+n+1}},1/2\,{\frac {4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) {y}^{2}{m}^{2}n+12\,{{\rm e}^{-3\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}}}}{27}^{-{\frac {m}{2\,m+n+1}}}{9}^{-{\frac {n+1}{2\,m+n+1}}}{2}^{-{\frac {m+n+1}{2\,m+n+1}}} \left ( {\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) ^{-{\frac {m+n+1}{2\,m+n+1}}}{x}^{-m} \WhittakerM \left ( {\frac {m+n+1}{2\,m+n+1}},1/2\,{\frac {4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) ybn+16\,{{\rm e}^{-3\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}}}}{27}^{-{\frac {m}{2\,m+n+1}}}{9}^{-{\frac {n+1}{2\,m+n+1}}}{2}^{-{\frac {m+n+1}{2\,m+n+1}}} \left ( {\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) ^{-{\frac {m+n+1}{2\,m+n+1}}}{x}^{-2\,m} \WhittakerM \left ( {\frac {m+n+1}{2\,m+n+1}},1/2\,{\frac {4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) {y}^{2}mn+16\,{{\rm e}^{-3\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}}}}{27}^{-{\frac {m}{2\,m+n+1}}}{9}^{-{\frac {n+1}{2\,m+n+1}}}{2}^{-{\frac {m+n+1}{2\,m+n+1}}} \left ( {\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) ^{-{\frac {m+n+1}{2\,m+n+1}}}{x}^{-m} \WhittakerM \left ( -{\frac {m}{2\,m+n+1}},1/2\,{\frac {4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) yb{m}^{2}+{{\rm e}^{-3\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}}}}{2}^{{\frac {m}{2\,m+n+1}}}{27}^{-{\frac {m}{2\,m+n+1}}}{9}^{-{\frac {n+1}{2\,m+n+1}}} \left ( {\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) ^{-{\frac {m+n+1}{2\,m+n+1}}}{x}^{-m} \WhittakerM \left ( -{\frac {m}{2\,m+n+1}},1/2\,{\frac {4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) yb{n}^{3}+6\,{{\rm e}^{-3\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}}}}{27}^{-{\frac {m}{2\,m+n+1}}}{9}^{-{\frac {n+1}{2\,m+n+1}}}{2}^{-{\frac {m+n+1}{2\,m+n+1}}} \left ( {\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) ^{-{\frac {m+n+1}{2\,m+n+1}}}{x}^{-m} \WhittakerM \left ( -{\frac {m}{2\,m+n+1}},1/2\,{\frac {4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) yb{n}^{2}+20\,{{\rm e}^{-3\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}}}}{27}^{-{\frac {m}{2\,m+n+1}}}{9}^{-{\frac {n+1}{2\,m+n+1}}}{2}^{-{\frac {m+n+1}{2\,m+n+1}}} \left ( {\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) ^{-{\frac {m+n+1}{2\,m+n+1}}}{x}^{-m} \WhittakerM \left ( {\frac {m+n+1}{2\,m+n+1}},1/2\,{\frac {4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) yb{m}^{2}+8\,{{\rm e}^{-3\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}}}}{27}^{-{\frac {m}{2\,m+n+1}}}{9}^{-{\frac {n+1}{2\,m+n+1}}}{2}^{-{\frac {m+n+1}{2\,m+n+1}}} \left ( {\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) ^{-{\frac {m+n+1}{2\,m+n+1}}}{x}^{-m} \WhittakerM \left ( {\frac {m+n+1}{2\,m+n+1}},1/2\,{\frac {4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) yb{m}^{3}+6\,{{\rm e}^{-3\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}}}}{27}^{-{\frac {m}{2\,m+n+1}}}{9}^{-{\frac {n+1}{2\,m+n+1}}}{2}^{-{\frac {m+n+1}{2\,m+n+1}}} \left ( {\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) ^{-{\frac {m+n+1}{2\,m+n+1}}}{x}^{m+n+1} \WhittakerM \left ( -{\frac {m}{2\,m+n+1}},1/2\,{\frac {4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) ya{b}^{3}+4\,{{\rm e}^{-3\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}}}}{27}^{-{\frac {m}{2\,m+n+1}}}{9}^{-{\frac {n+1}{2\,m+n+1}}}{2}^{-{\frac {m+n+1}{2\,m+n+1}}} \left ( {\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) ^{-{\frac {m+n+1}{2\,m+n+1}}}{x}^{-m} \WhittakerM \left ( {\frac {m+n+1}{2\,m+n+1}},1/2\,{\frac {4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) yb{n}^{3}+12\,{{\rm e}^{-3\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}}}}{27}^{-{\frac {m}{2\,m+n+1}}}{9}^{-{\frac {n+1}{2\,m+n+1}}}{2}^{-{\frac {m+n+1}{2\,m+n+1}}} \left ( {\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) ^{-{\frac {m+n+1}{2\,m+n+1}}}{x}^{-m} \WhittakerM \left ( {\frac {m+n+1}{2\,m+n+1}},1/2\,{\frac {4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) yb{n}^{2}+10\,{{\rm e}^{-3\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}}}}{27}^{-{\frac {m}{2\,m+n+1}}}{9}^{-{\frac {n+1}{2\,m+n+1}}}{2}^{-{\frac {m+n+1}{2\,m+n+1}}} \left ( {\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) ^{-{\frac {m+n+1}{2\,m+n+1}}}{x}^{-m} \WhittakerM \left ( -{\frac {m}{2\,m+n+1}},1/2\,{\frac {4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) ybm+6\,{{\rm e}^{-3\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}}}}{27}^{-{\frac {m}{2\,m+n+1}}}{9}^{-{\frac {n+1}{2\,m+n+1}}}{2}^{-{\frac {m+n+1}{2\,m+n+1}}} \left ( {\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) ^{-{\frac {m+n+1}{2\,m+n+1}}}{x}^{-m} \WhittakerM \left ( -{\frac {m}{2\,m+n+1}},1/2\,{\frac {4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) ybn+{{\rm e}^{-3\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}}}}{2}^{{\frac {m}{2\,m+n+1}}}{27}^{-{\frac {m}{2\,m+n+1}}}{9}^{-{\frac {n+1}{2\,m+n+1}}} \left ( {\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) ^{-{\frac {m+n+1}{2\,m+n+1}}} \WhittakerM \left ( {\frac {m+n+1}{2\,m+n+1}},1/2\,{\frac {4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) {b}^{2}+{{\rm e}^{-3\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}}}}{27}^{-{\frac {m}{2\,m+n+1}}}{9}^{-{\frac {n+1}{2\,m+n+1}}}{2}^{-{\frac {m+n+1}{2\,m+n+1}}} \left ( {\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) ^{-{\frac {m+n+1}{2\,m+n+1}}} \WhittakerM \left ( -{\frac {m}{2\,m+n+1}},1/2\,{\frac {4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) {b}^{2}+12\,{{\rm e}^{-3\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}}}}{27}^{-{\frac {m}{2\,m+n+1}}}{9}^{-{\frac {n+1}{2\,m+n+1}}}{2}^{-{\frac {m+n+1}{2\,m+n+1}}} \left ( {\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) ^{-{\frac {m+n+1}{2\,m+n+1}}}{x}^{m+n+1} \WhittakerM \left ( -{\frac {m}{2\,m+n+1}},1/2\,{\frac {4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) ya{b}^{3}n+12\,{{\rm e}^{-3\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}}}}{27}^{-{\frac {m}{2\,m+n+1}}}{9}^{-{\frac {n+1}{2\,m+n+1}}}{2}^{-{\frac {m+n+1}{2\,m+n+1}}} \left ( {\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) ^{-{\frac {m+n+1}{2\,m+n+1}}}{x}^{2\,m+n+1} \WhittakerM \left ( -{\frac {m}{2\,m+n+1}},1/2\,{\frac {4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) a{b}^{4}mn+12\,{{\rm e}^{-6\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}}}}{9}^{-{\frac {m}{2\,m+n+1}}}{3}^{-{\frac {n+1}{2\,m+n+1}}}{b}^{2}{m}^{2}+{{\rm e}^{-6\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}}}}{9}^{{\frac {m+n+1}{2\,m+n+1}}}{3}^{-{\frac {n+1}{2\,m+n+1}}}{b}^{2}{n}^{3}+27\,{{\rm e}^{-6\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}}}}{9}^{-{\frac {m}{2\,m+n+1}}}{3}^{-{\frac {n+1}{2\,m+n+1}}}{b}^{2}{n}^{2}+21\,{{\rm e}^{-6\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}}}}{9}^{-{\frac {m}{2\,m+n+1}}}{3}^{-{\frac {n+1}{2\,m+n+1}}}{b}^{2}m+27\,{{\rm e}^{-6\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}}}}{9}^{-{\frac {m}{2\,m+n+1}}}{3}^{-{\frac {n+1}{2\,m+n+1}}}{b}^{2}n+4\,{{\rm e}^{-3\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}}}}{27}^{-{\frac {m}{2\,m+n+1}}}{9}^{-{\frac {n+1}{2\,m+n+1}}}{2}^{-{\frac {m+n+1}{2\,m+n+1}}} \left ( {\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) ^{-{\frac {m+n+1}{2\,m+n+1}}}{x}^{-2\,m} \WhittakerM \left ( {\frac {m+n+1}{2\,m+n+1}},1/2\,{\frac {4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) {y}^{2}{m}^{3}+10\,{{\rm e}^{-3\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}}}}{27}^{-{\frac {m}{2\,m+n+1}}}{9}^{-{\frac {n+1}{2\,m+n+1}}}{2}^{-{\frac {m+n+1}{2\,m+n+1}}} \left ( {\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) ^{-{\frac {m+n+1}{2\,m+n+1}}}{x}^{-2\,m} \WhittakerM \left ( {\frac {m+n+1}{2\,m+n+1}},1/2\,{\frac {4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) {y}^{2}{m}^{2}+{{\rm e}^{-3\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}}}}{2}^{{\frac {m}{2\,m+n+1}}}{27}^{-{\frac {m}{2\,m+n+1}}}{9}^{-{\frac {n+1}{2\,m+n+1}}} \left ( {\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) ^{-{\frac {m+n+1}{2\,m+n+1}}}{x}^{-2\,m} \WhittakerM \left ( {\frac {m+n+1}{2\,m+n+1}},1/2\,{\frac {4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) {y}^{2}{n}^{3}+4\,{{\rm e}^{-3\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}}}}{27}^{-{\frac {m}{2\,m+n+1}}}{9}^{-{\frac {n+1}{2\,m+n+1}}}{2}^{-{\frac {m+n+1}{2\,m+n+1}}} \left ( {\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) ^{-{\frac {m+n+1}{2\,m+n+1}}}{x}^{-m} \WhittakerM \left ( {\frac {m+n+1}{2\,m+n+1}},1/2\,{\frac {4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) yb+5\,{{\rm e}^{-3\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}}}}{27}^{-{\frac {m}{2\,m+n+1}}}{9}^{-{\frac {n+1}{2\,m+n+1}}}{2}^{-{\frac {m+n+1}{2\,m+n+1}}} \left ( {\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) ^{-{\frac {m+n+1}{2\,m+n+1}}}{x}^{-2\,m} \WhittakerM \left ( -{\frac {m}{2\,m+n+1}},1/2\,{\frac {4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) {y}^{2}m+6\,{{\rm e}^{-3\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}}}}{27}^{-{\frac {m}{2\,m+n+1}}}{9}^{-{\frac {n+1}{2\,m+n+1}}}{2}^{-{\frac {m+n+1}{2\,m+n+1}}} \left ( {\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) ^{-{\frac {m+n+1}{2\,m+n+1}}}{x}^{-2\,m} \WhittakerM \left ( {\frac {m+n+1}{2\,m+n+1}},1/2\,{\frac {4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) {y}^{2}{n}^{2}+{{\rm e}^{-3\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}}}}{2}^{{\frac {m}{2\,m+n+1}}}{27}^{-{\frac {m}{2\,m+n+1}}}{9}^{-{\frac {n+1}{2\,m+n+1}}} \left ( {\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) ^{-{\frac {m+n+1}{2\,m+n+1}}}{x}^{-m} \WhittakerM \left ( -{\frac {m}{2\,m+n+1}},1/2\,{\frac {4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) yb+10\,{{\rm e}^{-3\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}}}}{27}^{-{\frac {m}{2\,m+n+1}}}{9}^{-{\frac {n+1}{2\,m+n+1}}}{2}^{-{\frac {m+n+1}{2\,m+n+1}}} \left ( {\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) ^{-{\frac {m+n+1}{2\,m+n+1}}} \WhittakerM \left ( -{\frac {m}{2\,m+n+1}},1/2\,{\frac {4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) {b}^{2}mn+5\,{{\rm e}^{-3\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}}}}{27}^{-{\frac {m}{2\,m+n+1}}}{9}^{-{\frac {n+1}{2\,m+n+1}}}{2}^{-{\frac {m+n+1}{2\,m+n+1}}} \left ( {\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) ^{-{\frac {m+n+1}{2\,m+n+1}}} \WhittakerM \left ( -{\frac {m}{2\,m+n+1}},1/2\,{\frac {4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) {b}^{2}m{n}^{2}+3\,{{\rm e}^{-3\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}}}}{27}^{-{\frac {m}{2\,m+n+1}}}{9}^{-{\frac {n+1}{2\,m+n+1}}}{2}^{-{\frac {m+n+1}{2\,m+n+1}}} \left ( {\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) ^{-{\frac {m+n+1}{2\,m+n+1}}}{x}^{2\,m+n+1} \WhittakerM \left ( -{\frac {m}{2\,m+n+1}},1/2\,{\frac {4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) a{b}^{4}{n}^{2}+12\,{{\rm e}^{-3\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}}}}{27}^{-{\frac {m}{2\,m+n+1}}}{9}^{-{\frac {n+1}{2\,m+n+1}}}{2}^{-{\frac {m+n+1}{2\,m+n+1}}} \left ( {\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) ^{-{\frac {m+n+1}{2\,m+n+1}}}{x}^{2\,m+n+1} \WhittakerM \left ( -{\frac {m}{2\,m+n+1}},1/2\,{\frac {4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) a{b}^{4}m+6\,{{\rm e}^{-3\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}}}}{27}^{-{\frac {m}{2\,m+n+1}}}{9}^{-{\frac {n+1}{2\,m+n+1}}}{2}^{-{\frac {m+n+1}{2\,m+n+1}}} \left ( {\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) ^{-{\frac {m+n+1}{2\,m+n+1}}}{x}^{2\,m+n+1} \WhittakerM \left ( -{\frac {m}{2\,m+n+1}},1/2\,{\frac {4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) a{b}^{4}n+16\,{{\rm e}^{-3\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}}}}{27}^{-{\frac {m}{2\,m+n+1}}}{9}^{-{\frac {n+1}{2\,m+n+1}}}{2}^{-{\frac {m+n+1}{2\,m+n+1}}} \left ( {\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) ^{-{\frac {m+n+1}{2\,m+n+1}}}{x}^{-m} \WhittakerM \left ( -{\frac {m}{2\,m+n+1}},1/2\,{\frac {4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) yb{m}^{2}n+32\,{{\rm e}^{-3\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}}}}{27}^{-{\frac {m}{2\,m+n+1}}}{9}^{-{\frac {n+1}{2\,m+n+1}}}{2}^{-{\frac {m+n+1}{2\,m+n+1}}} \left ( {\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) ^{-{\frac {m+n+1}{2\,m+n+1}}}{x}^{-m} \WhittakerM \left ( {\frac {m+n+1}{2\,m+n+1}},1/2\,{\frac {4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) ybmn+10\,{{\rm e}^{-3\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}}}}{27}^{-{\frac {m}{2\,m+n+1}}}{9}^{-{\frac {n+1}{2\,m+n+1}}}{2}^{-{\frac {m+n+1}{2\,m+n+1}}} \left ( {\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) ^{-{\frac {m+n+1}{2\,m+n+1}}}{x}^{-m} \WhittakerM \left ( -{\frac {m}{2\,m+n+1}},1/2\,{\frac {4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) ybm{n}^{2}+20\,{{\rm e}^{-3\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}}}}{27}^{-{\frac {m}{2\,m+n+1}}}{9}^{-{\frac {n+1}{2\,m+n+1}}}{2}^{-{\frac {m+n+1}{2\,m+n+1}}} \left ( {\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) ^{-{\frac {m+n+1}{2\,m+n+1}}}{x}^{-m} \WhittakerM \left ( {\frac {m+n+1}{2\,m+n+1}},1/2\,{\frac {4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) yb{m}^{2}n+16\,{{\rm e}^{-3\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}}}}{27}^{-{\frac {m}{2\,m+n+1}}}{9}^{-{\frac {n+1}{2\,m+n+1}}}{2}^{-{\frac {m+n+1}{2\,m+n+1}}} \left ( {\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) ^{-{\frac {m+n+1}{2\,m+n+1}}}{x}^{-m} \WhittakerM \left ( {\frac {m+n+1}{2\,m+n+1}},1/2\,{\frac {4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) ybm{n}^{2}+20\,{{\rm e}^{-3\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}}}}{27}^{-{\frac {m}{2\,m+n+1}}}{9}^{-{\frac {n+1}{2\,m+n+1}}}{2}^{-{\frac {m+n+1}{2\,m+n+1}}} \left ( {\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) ^{-{\frac {m+n+1}{2\,m+n+1}}}{x}^{-m} \WhittakerM \left ( -{\frac {m}{2\,m+n+1}},1/2\,{\frac {4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) ybmn+24\,{{\rm e}^{-3\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}}}}{27}^{-{\frac {m}{2\,m+n+1}}}{9}^{-{\frac {n+1}{2\,m+n+1}}}{2}^{-{\frac {m+n+1}{2\,m+n+1}}} \left ( {\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) ^{-{\frac {m+n+1}{2\,m+n+1}}}{x}^{m+n+1} \WhittakerM \left ( -{\frac {m}{2\,m+n+1}},1/2\,{\frac {4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) ya{b}^{3}{m}^{2}+6\,{{\rm e}^{-3\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}}}}{27}^{-{\frac {m}{2\,m+n+1}}}{9}^{-{\frac {n+1}{2\,m+n+1}}}{2}^{-{\frac {m+n+1}{2\,m+n+1}}} \left ( {\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) ^{-{\frac {m+n+1}{2\,m+n+1}}}{x}^{m+n+1} \WhittakerM \left ( -{\frac {m}{2\,m+n+1}},1/2\,{\frac {4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) ya{b}^{3}{n}^{2}+12\,{{\rm e}^{-3\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}}}}{27}^{-{\frac {m}{2\,m+n+1}}}{9}^{-{\frac {n+1}{2\,m+n+1}}}{2}^{-{\frac {m+n+1}{2\,m+n+1}}} \left ( {\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) ^{-{\frac {m+n+1}{2\,m+n+1}}}{x}^{n+1} \WhittakerM \left ( -{\frac {m}{2\,m+n+1}},1/2\,{\frac {4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) {y}^{2}a{b}^{2}{m}^{2}+12\,{{\rm e}^{-3\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}}}}{27}^{-{\frac {m}{2\,m+n+1}}}{9}^{-{\frac {n+1}{2\,m+n+1}}}{2}^{-{\frac {m+n+1}{2\,m+n+1}}} \left ( {\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) ^{-{\frac {m+n+1}{2\,m+n+1}}}{x}^{n+1} \WhittakerM \left ( -{\frac {m}{2\,m+n+1}},1/2\,{\frac {4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) {y}^{2}a{b}^{2}m+3\,{{\rm e}^{-3\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}}}}{27}^{-{\frac {m}{2\,m+n+1}}}{9}^{-{\frac {n+1}{2\,m+n+1}}}{2}^{-{\frac {m+n+1}{2\,m+n+1}}} \left ( {\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) ^{-{\frac {m+n+1}{2\,m+n+1}}}{x}^{n+1} \WhittakerM \left ( -{\frac {m}{2\,m+n+1}},1/2\,{\frac {4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) {y}^{2}a{b}^{2}{n}^{2}+24\,{{\rm e}^{-3\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}}}}{27}^{-{\frac {m}{2\,m+n+1}}}{9}^{-{\frac {n+1}{2\,m+n+1}}}{2}^{-{\frac {m+n+1}{2\,m+n+1}}} \left ( {\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) ^{-{\frac {m+n+1}{2\,m+n+1}}}{x}^{m+n+1} \WhittakerM \left ( -{\frac {m}{2\,m+n+1}},1/2\,{\frac {4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) ya{b}^{3}m+10\,{{\rm e}^{-3\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}}}}{27}^{-{\frac {m}{2\,m+n+1}}}{9}^{-{\frac {n+1}{2\,m+n+1}}}{2}^{-{\frac {m+n+1}{2\,m+n+1}}} \left ( {\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) ^{-{\frac {m+n+1}{2\,m+n+1}}} \WhittakerM \left ( {\frac {m+n+1}{2\,m+n+1}},1/2\,{\frac {4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) {b}^{2}{m}^{2}+3\,{{\rm e}^{-3\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}}}}{27}^{-{\frac {m}{2\,m+n+1}}}{9}^{-{\frac {n+1}{2\,m+n+1}}}{2}^{-{\frac {m+n+1}{2\,m+n+1}}} \left ( {\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) ^{-{\frac {m+n+1}{2\,m+n+1}}} \WhittakerM \left ( -{\frac {m}{2\,m+n+1}},1/2\,{\frac {4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) {b}^{2}n+8\,{{\rm e}^{-3\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}}}}{27}^{-{\frac {m}{2\,m+n+1}}}{9}^{-{\frac {n+1}{2\,m+n+1}}}{2}^{-{\frac {m+n+1}{2\,m+n+1}}} \left ( {\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) ^{-{\frac {m+n+1}{2\,m+n+1}}} \WhittakerM \left ( {\frac {m+n+1}{2\,m+n+1}},1/2\,{\frac {4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) {b}^{2}m+6\,{{\rm e}^{-3\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}}}}{27}^{-{\frac {m}{2\,m+n+1}}}{9}^{-{\frac {n+1}{2\,m+n+1}}}{2}^{-{\frac {m+n+1}{2\,m+n+1}}} \left ( {\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) ^{-{\frac {m+n+1}{2\,m+n+1}}} \WhittakerM \left ( {\frac {m+n+1}{2\,m+n+1}},1/2\,{\frac {4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) {b}^{2}n+3\,{{\rm e}^{-3\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}}}}{27}^{-{\frac {m}{2\,m+n+1}}}{9}^{-{\frac {n+1}{2\,m+n+1}}}{2}^{-{\frac {m+n+1}{2\,m+n+1}}} \left ( {\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) ^{-{\frac {m+n+1}{2\,m+n+1}}} \WhittakerM \left ( -{\frac {m}{2\,m+n+1}},1/2\,{\frac {4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) {b}^{2}{n}^{2}+4\,{{\rm e}^{-3\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}}}}{27}^{-{\frac {m}{2\,m+n+1}}}{9}^{-{\frac {n+1}{2\,m+n+1}}}{2}^{-{\frac {m+n+1}{2\,m+n+1}}} \left ( {\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) ^{-{\frac {m+n+1}{2\,m+n+1}}} \WhittakerM \left ( {\frac {m+n+1}{2\,m+n+1}},1/2\,{\frac {4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) {b}^{2}{m}^{3}+8\,{{\rm e}^{-3\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}}}}{27}^{-{\frac {m}{2\,m+n+1}}}{9}^{-{\frac {n+1}{2\,m+n+1}}}{2}^{-{\frac {m+n+1}{2\,m+n+1}}} \left ( {\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) ^{-{\frac {m+n+1}{2\,m+n+1}}} \WhittakerM \left ( -{\frac {m}{2\,m+n+1}},1/2\,{\frac {4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) {b}^{2}{m}^{2}+{{\rm e}^{-3\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}}}}{2}^{{\frac {m}{2\,m+n+1}}}{27}^{-{\frac {m}{2\,m+n+1}}}{9}^{-{\frac {n+1}{2\,m+n+1}}} \left ( {\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) ^{-{\frac {m+n+1}{2\,m+n+1}}} \WhittakerM \left ( {\frac {m+n+1}{2\,m+n+1}},1/2\,{\frac {4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) {b}^{2}{n}^{3}+{{\rm e}^{-3\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}}}}{2}^{{\frac {m}{2\,m+n+1}}}{27}^{-{\frac {m}{2\,m+n+1}}}{9}^{-{\frac {n+1}{2\,m+n+1}}} \left ( {\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) ^{-{\frac {m+n+1}{2\,m+n+1}}}{x}^{-2\,m} \WhittakerM \left ( {\frac {m+n+1}{2\,m+n+1}},1/2\,{\frac {4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) {y}^{2}+{{\rm e}^{-3\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}}}}{27}^{-{\frac {m}{2\,m+n+1}}}{9}^{-{\frac {n+1}{2\,m+n+1}}}{2}^{-{\frac {m+n+1}{2\,m+n+1}}} \left ( {\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) ^{-{\frac {m+n+1}{2\,m+n+1}}}{x}^{-2\,m} \WhittakerM \left ( -{\frac {m}{2\,m+n+1}},1/2\,{\frac {4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) {y}^{2}+6\,{{\rm e}^{-3\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}}}}{27}^{-{\frac {m}{2\,m+n+1}}}{9}^{-{\frac {n+1}{2\,m+n+1}}}{2}^{-{\frac {m+n+1}{2\,m+n+1}}} \left ( {\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) ^{-{\frac {m+n+1}{2\,m+n+1}}} \WhittakerM \left ( {\frac {m+n+1}{2\,m+n+1}},1/2\,{\frac {4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) {b}^{2}{n}^{2}+5\,{{\rm e}^{-3\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}}}}{27}^{-{\frac {m}{2\,m+n+1}}}{9}^{-{\frac {n+1}{2\,m+n+1}}}{2}^{-{\frac {m+n+1}{2\,m+n+1}}} \left ( {\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) ^{-{\frac {m+n+1}{2\,m+n+1}}} \WhittakerM \left ( -{\frac {m}{2\,m+n+1}},1/2\,{\frac {4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) {b}^{2}m+4\,{{\rm e}^{-3\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}}}}{27}^{-{\frac {m}{2\,m+n+1}}}{9}^{-{\frac {n+1}{2\,m+n+1}}}{2}^{-{\frac {m+n+1}{2\,m+n+1}}} \left ( {\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) ^{-{\frac {m+n+1}{2\,m+n+1}}} \WhittakerM \left ( -{\frac {m}{2\,m+n+1}},1/2\,{\frac {4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) {b}^{2}{m}^{3}+{{\rm e}^{-3\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}}}}{27}^{-{\frac {m}{2\,m+n+1}}}{9}^{-{\frac {n+1}{2\,m+n+1}}}{2}^{-{\frac {m+n+1}{2\,m+n+1}}} \left ( {\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) ^{-{\frac {m+n+1}{2\,m+n+1}}} \WhittakerM \left ( -{\frac {m}{2\,m+n+1}},1/2\,{\frac {4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) {b}^{2}{n}^{3}+10\,{{\rm e}^{-3\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}}}}{27}^{-{\frac {m}{2\,m+n+1}}}{9}^{-{\frac {n+1}{2\,m+n+1}}}{2}^{-{\frac {m+n+1}{2\,m+n+1}}} \left ( {\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) ^{-{\frac {m+n+1}{2\,m+n+1}}} \WhittakerM \left ( {\frac {m+n+1}{2\,m+n+1}},1/2\,{\frac {4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) {b}^{2}{m}^{2}n+8\,{{\rm e}^{-3\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}}}}{27}^{-{\frac {m}{2\,m+n+1}}}{9}^{-{\frac {n+1}{2\,m+n+1}}}{2}^{-{\frac {m+n+1}{2\,m+n+1}}} \left ( {\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) ^{-{\frac {m+n+1}{2\,m+n+1}}} \WhittakerM \left ( -{\frac {m}{2\,m+n+1}},1/2\,{\frac {4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) {b}^{2}{m}^{2}n+3\,{{\rm e}^{-3\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}}}}{27}^{-{\frac {m}{2\,m+n+1}}}{9}^{-{\frac {n+1}{2\,m+n+1}}}{2}^{-{\frac {m+n+1}{2\,m+n+1}}} \left ( {\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) ^{-{\frac {m+n+1}{2\,m+n+1}}}{x}^{2\,m+n+1} \WhittakerM \left ( -{\frac {m}{2\,m+n+1}},1/2\,{\frac {4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) a{b}^{4}+16\,{{\rm e}^{-3\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}}}}{27}^{-{\frac {m}{2\,m+n+1}}}{9}^{-{\frac {n+1}{2\,m+n+1}}}{2}^{-{\frac {m+n+1}{2\,m+n+1}}} \left ( {\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) ^{-{\frac {m+n+1}{2\,m+n+1}}} \WhittakerM \left ( {\frac {m+n+1}{2\,m+n+1}},1/2\,{\frac {4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) {b}^{2}mn+8\,{{\rm e}^{-3\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}}}}{27}^{-{\frac {m}{2\,m+n+1}}}{9}^{-{\frac {n+1}{2\,m+n+1}}}{2}^{-{\frac {m+n+1}{2\,m+n+1}}} \left ( {\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) ^{-{\frac {m+n+1}{2\,m+n+1}}} \WhittakerM \left ( {\frac {m+n+1}{2\,m+n+1}},1/2\,{\frac {4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) {b}^{2}m{n}^{2}+3\,{{\rm e}^{-3\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}}}}{27}^{-{\frac {m}{2\,m+n+1}}}{9}^{-{\frac {n+1}{2\,m+n+1}}}{2}^{-{\frac {m+n+1}{2\,m+n+1}}} \left ( {\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) ^{-{\frac {m+n+1}{2\,m+n+1}}}{x}^{-2\,m} \WhittakerM \left ( -{\frac {m}{2\,m+n+1}},1/2\,{\frac {4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) {y}^{2}n+8\,{{\rm e}^{-3\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}}}}{27}^{-{\frac {m}{2\,m+n+1}}}{9}^{-{\frac {n+1}{2\,m+n+1}}}{2}^{-{\frac {m+n+1}{2\,m+n+1}}} \left ( {\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) ^{-{\frac {m+n+1}{2\,m+n+1}}}{x}^{-2\,m} \WhittakerM \left ( {\frac {m+n+1}{2\,m+n+1}},1/2\,{\frac {4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) {y}^{2}m+3\,{{\rm e}^{-3\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}}}}{27}^{-{\frac {m}{2\,m+n+1}}}{9}^{-{\frac {n+1}{2\,m+n+1}}}{2}^{-{\frac {m+n+1}{2\,m+n+1}}} \left ( {\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) ^{-{\frac {m+n+1}{2\,m+n+1}}}{x}^{-2\,m} \WhittakerM \left ( -{\frac {m}{2\,m+n+1}},1/2\,{\frac {4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) {y}^{2}{n}^{2}+6\,{{\rm e}^{-3\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}}}}{27}^{-{\frac {m}{2\,m+n+1}}}{9}^{-{\frac {n+1}{2\,m+n+1}}}{2}^{-{\frac {m+n+1}{2\,m+n+1}}} \left ( {\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) ^{-{\frac {m+n+1}{2\,m+n+1}}}{x}^{-2\,m} \WhittakerM \left ( {\frac {m+n+1}{2\,m+n+1}},1/2\,{\frac {4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) {y}^{2}n+8\,{{\rm e}^{-3\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}}}}{27}^{-{\frac {m}{2\,m+n+1}}}{9}^{-{\frac {n+1}{2\,m+n+1}}}{2}^{-{\frac {m+n+1}{2\,m+n+1}}} \left ( {\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) ^{-{\frac {m+n+1}{2\,m+n+1}}}{x}^{-2\,m} \WhittakerM \left ( -{\frac {m}{2\,m+n+1}},1/2\,{\frac {4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) {y}^{2}{m}^{2}+4\,{{\rm e}^{-3\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}}}}{27}^{-{\frac {m}{2\,m+n+1}}}{9}^{-{\frac {n+1}{2\,m+n+1}}}{2}^{-{\frac {m+n+1}{2\,m+n+1}}} \left ( {\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) ^{-{\frac {m+n+1}{2\,m+n+1}}}{x}^{-2\,m} \WhittakerM \left ( -{\frac {m}{2\,m+n+1}},1/2\,{\frac {4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) {y}^{2}{m}^{3}+{{\rm e}^{-3\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}}}}{27}^{-{\frac {m}{2\,m+n+1}}}{9}^{-{\frac {n+1}{2\,m+n+1}}}{2}^{-{\frac {m+n+1}{2\,m+n+1}}} \left ( {\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) ^{-{\frac {m+n+1}{2\,m+n+1}}}{x}^{-2\,m} \WhittakerM \left ( -{\frac {m}{2\,m+n+1}},1/2\,{\frac {4\,m+3\,n+3}{2\,m+n+1}},6\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}} \right ) {y}^{2}{n}^{3}+{{\rm e}^{-6\,{\frac {a{b}^{2}{x}^{2\,m+n+1}}{2\,m+n+1}}}}{9}^{{\frac {m+n+1}{2\,m+n+1}}}{3}^{-{\frac {n+1}{2\,m+n+1}}}{b}^{2} \right ) } \right ) \]
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Added January 2, 2019.
Problem 2.2.5.16 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + \left ( a x^n y^3 + 3 a b x^{n+m} y^2+ c x^k y- 2 a b^3 x^{n+3 m} + b c x^{m+l} - b m x^{m-1} \right ) w_y = 0 \]
Mathematica ✗
ClearAll[w, x, y, n, a, b, m, c, k]; pde = D[w[x, y], x] + (a*x^n*y^3 + 3*a*b*x^(n + m)*y^2 + c*x^k*y - 2*a*b^3*x^(n + 3*m) + b*c*x^(m + l) - b*m*x^(m - 1))*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \text {Failed} \]
Maple ✗
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k'; pde := diff(w(x,y),x)+ (a*x^n*y^3 + 3*a*b*x^(n+m)*y^2+ c*x^k*y-2*a*b^3*x^(n+3*m) + b*c*x^(m+l)-b*m*x^(m-1))*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ \text { sol=() } \]
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Added January 2, 2019.
Problem 2.2.5.17 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + \left ( a y^n + b x ^{\frac {n}{1-n}} \right ) w_y = 0 \]
Mathematica ✗
ClearAll[w, x, y, n, a, b, m, c, k]; pde = D[w[x, y], x] + (a*y^n + b*x^(n/(1 - n)))*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \text {Failed} \]
Maple ✓
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k'; pde := diff(w(x,y),x)+ (a*y^n+b*x^(n/(1-n)))*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( -\int _{{\it \_b}}^{y}\!{1{x}^{{\frac {n}{n-1}}} \left ( {{\it \_a}}^{n}{x}^{{\frac {n}{n-1}}}anx-a{{\it \_a}}^{n}{x}^{{\frac {n}{n-1}}}x+bxn+{x}^{{\frac {n}{n-1}}}{\it \_a}-bx \right ) ^{-1}}\,{\rm d}{\it \_a}n+\int _{{\it \_b}}^{y}\!{1{x}^{{\frac {n}{n-1}}} \left ( {{\it \_a}}^{n}{x}^{{\frac {n}{n-1}}}anx-a{{\it \_a}}^{n}{x}^{{\frac {n}{n-1}}}x+bxn+{x}^{{\frac {n}{n-1}}}{\it \_a}-bx \right ) ^{-1}}\,{\rm d}{\it \_a}+\ln \left ( x \right ) \right ) \]
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Added January 2, 2019.
Problem 2.2.5.18 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + \left ( a x^{m-n-(m n)} y^n + b x^m \right ) w_y = 0 \]
Mathematica ✗
ClearAll[w, x, y, n, a, b, m, c, k]; pde = D[w[x, y], x] + (a*x^(m - n - m*n)*y^n + b*x^m)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \text {Failed} \]
Maple ✓
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k'; pde := diff(w(x,y),x)+ (a*x^(m-n-(m*n))*y^n + b*x^m)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( \int _{{\it \_b}}^{y}\!-{\frac {{x}^{mn}{x}^{n}}{{x}^{m}bx{x}^{mn}{x}^{n}-{x}^{mn}{x}^{n}{\it \_a}\,m+{{\it \_a}}^{n}a{x}^{m}x-{x}^{mn}{x}^{n}{\it \_a}}}\,{\rm d}{\it \_a}+\ln \left ( x \right ) \right ) \]
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Added January 2, 2019.
Problem 2.2.5.19 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + \left ( a x^n y^k + b x^m y \right ) w_y = 0 \]
Mathematica ✗
ClearAll[w, x, y, n, a, b, m, c, k]; pde = D[w[x, y], x] + (a*x^n*y^k + b*x^m*y)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \text {\$Aborted} \] Timed out
Maple ✓
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k'; pde := diff(w(x,y),x)+ (a*x^n*y^k + b*x^m*y)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1}{b \left ( 2\,{m}^{2}n+3\,m{n}^{2}+{n}^{3}+2\,{m}^{2}+10\,mn+6\,{n}^{2}+7\,m+11\,n+6 \right ) } \left ( 10\,{{\rm e}^{{\frac {{x}^{m+1}b \left ( k-1 \right ) }{m+1}}}}{y}^{1-k}bmn-5\,{x}^{n-m} \left ( -{\frac {{x}^{m+1}b \left ( k-1 \right ) }{m+1}} \right ) ^{-1/2\,{\frac {m+n+2}{m+1}}}{{\rm e}^{1/2\,{\frac {{x}^{m+1}b \left ( k-1 \right ) }{m+1}}}} \WhittakerM \left ( -1/2\,{\frac {-n+m}{m+1}},1/2\,{\frac {2\,m+3+n}{m+1}},-{\frac {{x}^{m+1}b \left ( k-1 \right ) }{m+1}} \right ) am-4\,{x}^{n-m} \left ( -{\frac {{x}^{m+1}b \left ( k-1 \right ) }{m+1}} \right ) ^{-1/2\,{\frac {m+n+2}{m+1}}}{{\rm e}^{1/2\,{\frac {{x}^{m+1}b \left ( k-1 \right ) }{m+1}}}} \WhittakerM \left ( -1/2\,{\frac {-n+m}{m+1}},1/2\,{\frac {2\,m+3+n}{m+1}},-{\frac {{x}^{m+1}b \left ( k-1 \right ) }{m+1}} \right ) a{m}^{2}-{x}^{n-m} \left ( -{\frac {{x}^{m+1}b \left ( k-1 \right ) }{m+1}} \right ) ^{-1/2\,{\frac {m+n+2}{m+1}}}{{\rm e}^{1/2\,{\frac {{x}^{m+1}b \left ( k-1 \right ) }{m+1}}}} \WhittakerM \left ( -1/2\,{\frac {-n+m}{m+1}},1/2\,{\frac {2\,m+3+n}{m+1}},-{\frac {{x}^{m+1}b \left ( k-1 \right ) }{m+1}} \right ) a{m}^{3}-{x}^{n-m} \left ( -{\frac {{x}^{m+1}b \left ( k-1 \right ) }{m+1}} \right ) ^{-1/2\,{\frac {m+n+2}{m+1}}}{{\rm e}^{1/2\,{\frac {{x}^{m+1}b \left ( k-1 \right ) }{m+1}}}} \WhittakerM \left ( -1/2\,{\frac {-n+m}{m+1}},1/2\,{\frac {2\,m+3+n}{m+1}},-{\frac {{x}^{m+1}b \left ( k-1 \right ) }{m+1}} \right ) an-8\,{x}^{n-m} \left ( -{\frac {{x}^{m+1}b \left ( k-1 \right ) }{m+1}} \right ) ^{-1/2\,{\frac {m+n+2}{m+1}}}{{\rm e}^{1/2\,{\frac {{x}^{m+1}b \left ( k-1 \right ) }{m+1}}}} \WhittakerM \left ( 1/2\,{\frac {m+n+2}{m+1}},1/2\,{\frac {2\,m+3+n}{m+1}},-{\frac {{x}^{m+1}b \left ( k-1 \right ) }{m+1}} \right ) am-5\,{x}^{n-m} \left ( -{\frac {{x}^{m+1}b \left ( k-1 \right ) }{m+1}} \right ) ^{-1/2\,{\frac {m+n+2}{m+1}}}{{\rm e}^{1/2\,{\frac {{x}^{m+1}b \left ( k-1 \right ) }{m+1}}}} \WhittakerM \left ( 1/2\,{\frac {m+n+2}{m+1}},1/2\,{\frac {2\,m+3+n}{m+1}},-{\frac {{x}^{m+1}b \left ( k-1 \right ) }{m+1}} \right ) a{m}^{2}-{x}^{n-m} \left ( -{\frac {{x}^{m+1}b \left ( k-1 \right ) }{m+1}} \right ) ^{-1/2\,{\frac {m+n+2}{m+1}}}{{\rm e}^{1/2\,{\frac {{x}^{m+1}b \left ( k-1 \right ) }{m+1}}}} \WhittakerM \left ( 1/2\,{\frac {m+n+2}{m+1}},1/2\,{\frac {2\,m+3+n}{m+1}},-{\frac {{x}^{m+1}b \left ( k-1 \right ) }{m+1}} \right ) a{m}^{3}-4\,{x}^{n-m} \left ( -{\frac {{x}^{m+1}b \left ( k-1 \right ) }{m+1}} \right ) ^{-1/2\,{\frac {m+n+2}{m+1}}}{{\rm e}^{1/2\,{\frac {{x}^{m+1}b \left ( k-1 \right ) }{m+1}}}} \WhittakerM \left ( 1/2\,{\frac {m+n+2}{m+1}},1/2\,{\frac {2\,m+3+n}{m+1}},-{\frac {{x}^{m+1}b \left ( k-1 \right ) }{m+1}} \right ) an-{x}^{n-m} \left ( -{\frac {{x}^{m+1}b \left ( k-1 \right ) }{m+1}} \right ) ^{-1/2\,{\frac {m+n+2}{m+1}}}{{\rm e}^{1/2\,{\frac {{x}^{m+1}b \left ( k-1 \right ) }{m+1}}}} \WhittakerM \left ( 1/2\,{\frac {m+n+2}{m+1}},1/2\,{\frac {2\,m+3+n}{m+1}},-{\frac {{x}^{m+1}b \left ( k-1 \right ) }{m+1}} \right ) a{n}^{2}- \left ( -{\frac {{x}^{m+1}b \left ( k-1 \right ) }{m+1}} \right ) ^{-1/2\,{\frac {m+n+2}{m+1}}}{{\rm e}^{1/2\,{\frac {{x}^{m+1}b \left ( k-1 \right ) }{m+1}}}} \WhittakerM \left ( -1/2\,{\frac {-n+m}{m+1}},1/2\,{\frac {2\,m+3+n}{m+1}},-{\frac {{x}^{m+1}b \left ( k-1 \right ) }{m+1}} \right ) ab{x}^{n+1}+6\,{{\rm e}^{{\frac {{x}^{m+1}b \left ( k-1 \right ) }{m+1}}}}{y}^{1-k}b+2\, \left ( -{\frac {{x}^{m+1}b \left ( k-1 \right ) }{m+1}} \right ) ^{-1/2\,{\frac {m+n+2}{m+1}}}{{\rm e}^{1/2\,{\frac {{x}^{m+1}b \left ( k-1 \right ) }{m+1}}}} \WhittakerM \left ( -1/2\,{\frac {-n+m}{m+1}},1/2\,{\frac {2\,m+3+n}{m+1}},-{\frac {{x}^{m+1}b \left ( k-1 \right ) }{m+1}} \right ) abkm{x}^{n+1}+ \left ( -{\frac {{x}^{m+1}b \left ( k-1 \right ) }{m+1}} \right ) ^{-1/2\,{\frac {m+n+2}{m+1}}}{{\rm e}^{1/2\,{\frac {{x}^{m+1}b \left ( k-1 \right ) }{m+1}}}} \WhittakerM \left ( -1/2\,{\frac {-n+m}{m+1}},1/2\,{\frac {2\,m+3+n}{m+1}},-{\frac {{x}^{m+1}b \left ( k-1 \right ) }{m+1}} \right ) abk{m}^{2}{x}^{n+1}-2\,{x}^{n-m} \left ( -{\frac {{x}^{m+1}b \left ( k-1 \right ) }{m+1}} \right ) ^{-1/2\,{\frac {m+n+2}{m+1}}}{{\rm e}^{1/2\,{\frac {{x}^{m+1}b \left ( k-1 \right ) }{m+1}}}} \WhittakerM \left ( -1/2\,{\frac {-n+m}{m+1}},1/2\,{\frac {2\,m+3+n}{m+1}},-{\frac {{x}^{m+1}b \left ( k-1 \right ) }{m+1}} \right ) a+2\,{{\rm e}^{{\frac {{x}^{m+1}b \left ( k-1 \right ) }{m+1}}}}{y}^{1-k}b{m}^{2}n+3\,{{\rm e}^{{\frac {{x}^{m+1}b \left ( k-1 \right ) }{m+1}}}}{y}^{1-k}bm{n}^{2}+{{\rm e}^{{\frac {{x}^{m+1}b \left ( k-1 \right ) }{m+1}}}}{y}^{1-k}b{n}^{3}+11\,{{\rm e}^{{\frac {{x}^{m+1}b \left ( k-1 \right ) }{m+1}}}}{y}^{1-k}bn+2\,{{\rm e}^{{\frac {{x}^{m+1}b \left ( k-1 \right ) }{m+1}}}}{y}^{1-k}b{m}^{2}+6\,{{\rm e}^{{\frac {{x}^{m+1}b \left ( k-1 \right ) }{m+1}}}}{y}^{1-k}b{n}^{2}+7\,{{\rm e}^{{\frac {{x}^{m+1}b \left ( k-1 \right ) }{m+1}}}}{y}^{1-k}bm-4\,{x}^{n-m} \left ( -{\frac {{x}^{m+1}b \left ( k-1 \right ) }{m+1}} \right ) ^{-1/2\,{\frac {m+n+2}{m+1}}}{{\rm e}^{1/2\,{\frac {{x}^{m+1}b \left ( k-1 \right ) }{m+1}}}} \WhittakerM \left ( 1/2\,{\frac {m+n+2}{m+1}},1/2\,{\frac {2\,m+3+n}{m+1}},-{\frac {{x}^{m+1}b \left ( k-1 \right ) }{m+1}} \right ) a-2\, \left ( -{\frac {{x}^{m+1}b \left ( k-1 \right ) }{m+1}} \right ) ^{-1/2\,{\frac {m+n+2}{m+1}}}{{\rm e}^{1/2\,{\frac {{x}^{m+1}b \left ( k-1 \right ) }{m+1}}}} \WhittakerM \left ( -1/2\,{\frac {-n+m}{m+1}},1/2\,{\frac {2\,m+3+n}{m+1}},-{\frac {{x}^{m+1}b \left ( k-1 \right ) }{m+1}} \right ) abm{x}^{n+1}- \left ( -{\frac {{x}^{m+1}b \left ( k-1 \right ) }{m+1}} \right ) ^{-1/2\,{\frac {m+n+2}{m+1}}}{{\rm e}^{1/2\,{\frac {{x}^{m+1}b \left ( k-1 \right ) }{m+1}}}} \WhittakerM \left ( -1/2\,{\frac {-n+m}{m+1}},1/2\,{\frac {2\,m+3+n}{m+1}},-{\frac {{x}^{m+1}b \left ( k-1 \right ) }{m+1}} \right ) ab{m}^{2}{x}^{n+1}+ \left ( -{\frac {{x}^{m+1}b \left ( k-1 \right ) }{m+1}} \right ) ^{-1/2\,{\frac {m+n+2}{m+1}}}{{\rm e}^{1/2\,{\frac {{x}^{m+1}b \left ( k-1 \right ) }{m+1}}}} \WhittakerM \left ( -1/2\,{\frac {-n+m}{m+1}},1/2\,{\frac {2\,m+3+n}{m+1}},-{\frac {{x}^{m+1}b \left ( k-1 \right ) }{m+1}} \right ) abk{x}^{n+1}-2\,{x}^{n-m} \left ( -{\frac {{x}^{m+1}b \left ( k-1 \right ) }{m+1}} \right ) ^{-1/2\,{\frac {m+n+2}{m+1}}}{{\rm e}^{1/2\,{\frac {{x}^{m+1}b \left ( k-1 \right ) }{m+1}}}} \WhittakerM \left ( -1/2\,{\frac {-n+m}{m+1}},1/2\,{\frac {2\,m+3+n}{m+1}},-{\frac {{x}^{m+1}b \left ( k-1 \right ) }{m+1}} \right ) amn-{x}^{n-m} \left ( -{\frac {{x}^{m+1}b \left ( k-1 \right ) }{m+1}} \right ) ^{-1/2\,{\frac {m+n+2}{m+1}}}{{\rm e}^{1/2\,{\frac {{x}^{m+1}b \left ( k-1 \right ) }{m+1}}}} \WhittakerM \left ( -1/2\,{\frac {-n+m}{m+1}},1/2\,{\frac {2\,m+3+n}{m+1}},-{\frac {{x}^{m+1}b \left ( k-1 \right ) }{m+1}} \right ) a{m}^{2}n-6\,{x}^{n-m} \left ( -{\frac {{x}^{m+1}b \left ( k-1 \right ) }{m+1}} \right ) ^{-1/2\,{\frac {m+n+2}{m+1}}}{{\rm e}^{1/2\,{\frac {{x}^{m+1}b \left ( k-1 \right ) }{m+1}}}} \WhittakerM \left ( 1/2\,{\frac {m+n+2}{m+1}},1/2\,{\frac {2\,m+3+n}{m+1}},-{\frac {{x}^{m+1}b \left ( k-1 \right ) }{m+1}} \right ) amn-{x}^{n-m} \left ( -{\frac {{x}^{m+1}b \left ( k-1 \right ) }{m+1}} \right ) ^{-1/2\,{\frac {m+n+2}{m+1}}}{{\rm e}^{1/2\,{\frac {{x}^{m+1}b \left ( k-1 \right ) }{m+1}}}} \WhittakerM \left ( 1/2\,{\frac {m+n+2}{m+1}},1/2\,{\frac {2\,m+3+n}{m+1}},-{\frac {{x}^{m+1}b \left ( k-1 \right ) }{m+1}} \right ) am{n}^{2}-2\,{x}^{n-m} \left ( -{\frac {{x}^{m+1}b \left ( k-1 \right ) }{m+1}} \right ) ^{-1/2\,{\frac {m+n+2}{m+1}}}{{\rm e}^{1/2\,{\frac {{x}^{m+1}b \left ( k-1 \right ) }{m+1}}}} \WhittakerM \left ( 1/2\,{\frac {m+n+2}{m+1}},1/2\,{\frac {2\,m+3+n}{m+1}},-{\frac {{x}^{m+1}b \left ( k-1 \right ) }{m+1}} \right ) a{m}^{2}n \right ) } \right ) \]
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Added January 2, 2019.
Problem 2.2.5.20 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ x w_x + \left ( a y^2 + b y+ c x^{2 b} \right ) w_y = 0 \]
Mathematica ✓
ClearAll[w, x, y, n, a, b, m, c, k]; pde = x*D[w[x, y], x] + (a*y^2 + b*y + c*x^(2*b))*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to c_1\left (\frac {-\sqrt {a} y \sin \left (\frac {\sqrt {a} \sqrt {c} x^b}{b}\right )-\sqrt {c} x^b \cos \left (\frac {\sqrt {a} \sqrt {c} x^b}{b}\right )}{\sqrt {a} y \cos \left (\frac {\sqrt {a} \sqrt {c} x^b}{b}\right )-\sqrt {c} x^b \sin \left (\frac {\sqrt {a} \sqrt {c} x^b}{b}\right )}\right )\right \}\right \} \]
Maple ✓
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k'; pde := x*diff(w(x,y),x)+ (a*y^2 + b*y+ c*x^(2*b))*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1}{b} \left ( \sqrt {c}\sqrt {a}{x}^{b}-\arctan \left ( {\frac {\sqrt {a}{x}^{-b}y}{\sqrt {c}}} \right ) b \right ) } \right ) \]
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Added January 2, 2019.
Problem 2.2.5.21 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ x w_x + \left ( a y^2+(n+b x^n) y + c x^{2 n} \right ) w_y = 0 \]
Mathematica ✓
ClearAll[w, x, y, n, a, b, m, c, k]; pde = x*D[w[x, y], x] + (a*y^2 + (n + b*x^n)*y + c*x^(2*n))*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to c_1\left (\frac {\exp \left (\frac {\sqrt {a} \sqrt {c} x^n \left (\frac {\sqrt {b^2-4 a c}}{\sqrt {a} \sqrt {c}}+\frac {b}{\sqrt {a} \sqrt {c}}\right )}{2 n}-\frac {\sqrt {a} \sqrt {c} x^n \left (\frac {b}{\sqrt {a} \sqrt {c}}-\frac {\sqrt {b^2-4 a c}}{\sqrt {a} \sqrt {c}}\right )}{2 n}\right ) \left (x^n \sqrt {b^2-4 a c}+2 a y+b x^n\right )}{x^n \sqrt {b^2-4 a c}-2 a y-b x^n}\right )\right \}\right \} \]
Maple ✓
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k'; pde := x*diff(w(x,y),x)+ (a*y^2+(n+b*x^n)*y + c*x^(2*n))*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {b}{\sqrt {{b}^{2} \left ( 4\,ca-{b}^{2} \right ) }n} \left ( 2\,bn\arctan \left ( {\frac {b \left ( b+2\,{x}^{-n}ya \right ) }{\sqrt {{b}^{2} \left ( 4\,ca-{b}^{2} \right ) }}} \right ) -\sqrt {{b}^{2} \left ( 4\,ca-{b}^{2} \right ) }{x}^{n} \right ) } \right ) \]
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Added January 2, 2019.
Problem 2.2.5.22 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ x w_x + \left ( a x^n y^2 + b y+ c x^{-n} \right ) w_y = 0 \]
Mathematica ✓
ClearAll[w, x, y, n, a, b, m, c, k]; pde = x*D[w[x, y], x] + (a*x^n*y^2 + b*y + c/x^n)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to c_1\left (\frac {-2 a y x^{\frac {1}{2} \sqrt {a} \sqrt {c} \left (\frac {\sqrt {-4 a c+b^2+2 b n+n^2}}{\sqrt {a} \sqrt {c}}-\frac {-b-n}{\sqrt {a} \sqrt {c}}\right )+n}-b x^{\frac {1}{2} \sqrt {a} \sqrt {c} \left (\frac {\sqrt {-4 a c+b^2+2 b n+n^2}}{\sqrt {a} \sqrt {c}}-\frac {-b-n}{\sqrt {a} \sqrt {c}}\right )}-n x^{\frac {1}{2} \sqrt {a} \sqrt {c} \left (\frac {\sqrt {-4 a c+b^2+2 b n+n^2}}{\sqrt {a} \sqrt {c}}-\frac {-b-n}{\sqrt {a} \sqrt {c}}\right )}-\sqrt {-4 a c+b^2+2 b n+n^2} x^{\frac {1}{2} \sqrt {a} \sqrt {c} \left (\frac {\sqrt {-4 a c+b^2+2 b n+n^2}}{\sqrt {a} \sqrt {c}}-\frac {-b-n}{\sqrt {a} \sqrt {c}}\right )}}{2 a y x^{\frac {1}{2} \sqrt {a} \sqrt {c} \left (-\frac {\sqrt {-4 a c+b^2+2 b n+n^2}}{\sqrt {a} \sqrt {c}}-\frac {-b-n}{\sqrt {a} \sqrt {c}}\right )+n}+b x^{\frac {1}{2} \sqrt {a} \sqrt {c} \left (-\frac {\sqrt {-4 a c+b^2+2 b n+n^2}}{\sqrt {a} \sqrt {c}}-\frac {-b-n}{\sqrt {a} \sqrt {c}}\right )}+n x^{\frac {1}{2} \sqrt {a} \sqrt {c} \left (-\frac {\sqrt {-4 a c+b^2+2 b n+n^2}}{\sqrt {a} \sqrt {c}}-\frac {-b-n}{\sqrt {a} \sqrt {c}}\right )}-\sqrt {-4 a c+b^2+2 b n+n^2} x^{\frac {1}{2} \sqrt {a} \sqrt {c} \left (-\frac {\sqrt {-4 a c+b^2+2 b n+n^2}}{\sqrt {a} \sqrt {c}}-\frac {-b-n}{\sqrt {a} \sqrt {c}}\right )}}\right )\right \}\right \} \]
Maple ✓
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k'; pde := x*diff(w(x,y),x)+ (a*x^n*y^2+b*y+c*x^(-n))*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1}{\sqrt {4\,ca-{b}^{2}-2\,bn-{n}^{2}}} \left ( \ln \left ( x \right ) \sqrt {4\,ca-{b}^{2}-2\,bn-{n}^{2}}-2\,\arctan \left ( {\frac {2\,a{x}^{n}y+b+n}{\sqrt {4\,ca-{b}^{2}-2\,bn-{n}^{2}}}} \right ) \right ) } \right ) \]
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Added January 2, 2019.
Problem 2.2.5.23 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ x w_x + \left ( a x^n y^2+ m y- a b^2 x^{x+2 m} \right ) w_y = 0 \]
Mathematica ✗
ClearAll[w, x, y, n, a, b, m, c, k]; pde = x*D[w[x, y], x] + (a*x^n*y^2 + m*y - a*b^2*x^(x + 2*m))*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \text {Failed} \]
Maple ✓
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k'; pde := x*diff(w(x,y),x)+ (a*x^n*y^2+ m*y- a*b^2*x^(x+2*m))*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{1 \left ( 2\,\BesselI \left ( -{\frac {n+m}{n+x+2\,m}},2\,{\frac {ab{x}^{n/2}{x}^{x/2}{x}^{m}}{n+x+2\,m}} \right ) {x}^{n}ya+2\,x{\frac {\partial }{\partial x}}\BesselI \left ( -{\frac {n+m}{n+x+2\,m}},2\,{\frac {ab{x}^{n/2}{x}^{x/2}{x}^{m}}{n+x+2\,m}} \right ) +\BesselI \left ( -{\frac {n+m}{n+x+2\,m}},2\,{\frac {ab{x}^{n/2}{x}^{x/2}{x}^{m}}{n+x+2\,m}} \right ) m+\BesselI \left ( -{\frac {n+m}{n+x+2\,m}},2\,{\frac {ab{x}^{n/2}{x}^{x/2}{x}^{m}}{n+x+2\,m}} \right ) n \right ) \left ( 2\,\BesselK \left ( {\frac {n+m}{n+x+2\,m}},2\,{\frac {ab{x}^{n/2}{x}^{x/2}{x}^{m}}{n+x+2\,m}} \right ) {x}^{n}ya+2\,x{\frac {\partial }{\partial x}}\BesselK \left ( {\frac {n+m}{n+x+2\,m}},2\,{\frac {ab{x}^{n/2}{x}^{x/2}{x}^{m}}{n+x+2\,m}} \right ) +\BesselK \left ( {\frac {n+m}{n+x+2\,m}},2\,{\frac {ab{x}^{n/2}{x}^{x/2}{x}^{m}}{n+x+2\,m}} \right ) m+\BesselK \left ( {\frac {n+m}{n+x+2\,m}},2\,{\frac {ab{x}^{n/2}{x}^{x/2}{x}^{m}}{n+x+2\,m}} \right ) n \right ) ^{-1}} \right ) \]
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Added January 2, 2019.
Problem 2.2.5.24 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ x w_x + \left ( x^{2 n} y^2+(m-n) y+ x^{2 m} \right ) w_y = 0 \]
Mathematica ✓
ClearAll[w, x, y, n, a, b, m, c, k]; pde = x*D[w[x, y], x] + (x^(2*n)*y^2 + (m - n)*y + x^(2*m))*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to c_1\left (\frac {m \tan ^{-1}\left (y x^{n-m}\right )+n \tan ^{-1}\left (y x^{n-m}\right )-x^{m+n}}{m+n}\right )\right \}\right \} \]
Maple ✓
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k'; pde := x*diff(w(x,y),x)+ (x^(2*n)*y^2+(m-n)*y+ x^(2*m))*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {-\arctan \left ( {x}^{n-m}y \right ) m-\arctan \left ( {x}^{n-m}y \right ) n+{x}^{n+m}}{n+m}} \right ) \]
____________________________________________________________________________________
Added January 2, 2019.
Problem 2.2.5.25 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ x w_x + \left ( a x^{2 n} y^2+ (b x^n -n) y + c \right ) w_y = 0 \]
Mathematica ✓
ClearAll[w, x, y, n, a, b, m, c, k]; pde = x*D[w[x, y], x] + (a*x^(2*n)*y^2 + (b*x^n - n)*y + c)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to c_1\left (\frac {\exp \left (\frac {\sqrt {a} \sqrt {c} x^n \left (\frac {\sqrt {b^2-4 a c}}{\sqrt {a} \sqrt {c}}+\frac {b}{\sqrt {a} \sqrt {c}}\right )}{2 n}-\frac {\sqrt {a} \sqrt {c} x^n \left (\frac {b}{\sqrt {a} \sqrt {c}}-\frac {\sqrt {b^2-4 a c}}{\sqrt {a} \sqrt {c}}\right )}{2 n}\right ) \left (\sqrt {b^2-4 a c}+2 a y x^n+b\right )}{\sqrt {b^2-4 a c}-2 a y x^n-b}\right )\right \}\right \} \]
Maple ✓
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k'; pde := x*diff(w(x,y),x)+ (a*x^(2*n)*y^2+ (b*x^n -n)*y + c)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {b}{\sqrt {{b}^{2} \left ( 4\,ca-{b}^{2} \right ) }n} \left ( 2\,bn\arctan \left ( {\frac {b \left ( 2\,a{x}^{n}y+b \right ) }{\sqrt {4\,ac{b}^{2}-{b}^{4}}}} \right ) -\sqrt {{b}^{2} \left ( 4\,ca-{b}^{2} \right ) }{x}^{n} \right ) } \right ) \]
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Added January 2, 2019.
Problem 2.2.5.26 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ x w_x + \left ( a x^{2 n + m} y^2 +(b x^{n+m}-n) y+ c x^m \right ) w_y = 0 \]
Mathematica ✓
ClearAll[w, x, y, n, a, b, m, c, k]; pde = x*D[w[x, y], x] + (a*x^(2*n + m)*y^2 + (b*x^(n + m) - n)*y + c*x^m)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to c_1\left (\frac {\left (\sqrt {b^2-4 a c}+2 a y x^n+b\right ) \exp \left (\frac {\sqrt {a} \sqrt {c} \left (\frac {\sqrt {b^2-4 a c}}{\sqrt {a} \sqrt {c}}+\frac {b}{\sqrt {a} \sqrt {c}}\right ) x^{m+n}}{2 (m+n)}-\frac {\sqrt {a} \sqrt {c} \left (\frac {b}{\sqrt {a} \sqrt {c}}-\frac {\sqrt {b^2-4 a c}}{\sqrt {a} \sqrt {c}}\right ) x^{m+n}}{2 (m+n)}\right )}{\sqrt {b^2-4 a c}-2 a y x^n-b}\right )\right \}\right \} \]
Maple ✓
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k'; pde := x*diff(w(x,y),x)+ (a*x^(2*n + m)*y^2 +(b*x^(n+m)-n)*y+ c*x^m)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {b}{\sqrt {{b}^{2} \left ( 4\,ca-{b}^{2} \right ) } \left ( n+m \right ) } \left ( 2\,bm\arctan \left ( {\frac {b \left ( 2\,a{x}^{n}y+b \right ) }{\sqrt {4\,ac{b}^{2}-{b}^{4}}}} \right ) +2\,bn\arctan \left ( {\frac {b \left ( 2\,a{x}^{n}y+b \right ) }{\sqrt {4\,ac{b}^{2}-{b}^{4}}}} \right ) -\sqrt {{b}^{2} \left ( 4\,ca-{b}^{2} \right ) }{x}^{n}{x}^{m} \right ) } \right ) \]
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Added January 2, 2019.
Problem 2.2.5.27 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ x w_x + \left ( a y^3+3 a b x^n y^2 - b n x^n -2 a b^3 x^{3 n} \right ) w_y = 0 \]
Mathematica ✓
ClearAll[w, x, y, n, a, b, m, c, k]; pde = x*D[w[x, y], x] + (a*y^3 + 3*a*b*x^n*y^2 - b*n*x^n - 2*a*b^3*x^(3*n))*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to c_1\left (\frac {e^{-\frac {3 a b^2 x^{2 n}}{n}} \left (a y^2 e^{\frac {3 a b^2 x^{2 n}}{n}} \text {ExpIntegralEi}\left (-\frac {3 a b^2 x^{2 n}}{n}\right )+a b^2 x^{2 n} e^{\frac {3 a b^2 x^{2 n}}{n}} \text {ExpIntegralEi}\left (-\frac {3 a b^2 x^{2 n}}{n}\right )+2 a b y x^n e^{\frac {3 a b^2 x^{2 n}}{n}} \text {ExpIntegralEi}\left (-\frac {3 a b^2 x^{2 n}}{n}\right )+n\right )}{n \left (b x^n+y\right )^2}\right )\right \}\right \} \]
Maple ✓
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k'; pde := x*diff(w(x,y),x)+ (a*y^3+3*a*b*x^n*y^2 - b*n*x^n -2*a*b^3*x^(3*n) )*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac {1}{n \left ( {x}^{2\,n}{b}^{2}+2\,{x}^{n}by+{y}^{2} \right ) } \left ( \Ei \left ( 1,3\,{\frac {{x}^{2\,n}a{b}^{2}}{n}} \right ) {x}^{2\,n}a{b}^{2}+2\,\Ei \left ( 1,3\,{\frac {{x}^{2\,n}a{b}^{2}}{n}} \right ) {x}^{n}yab+\Ei \left ( 1,3\,{\frac {{x}^{2\,n}a{b}^{2}}{n}} \right ) {y}^{2}a-{{\rm e}^{-3\,{\frac {{x}^{2\,n}a{b}^{2}}{n}}}}n \right ) } \right ) \]
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Added January 2, 2019.
Problem 2.2.5.28 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ x w_x + \left ( a x^{2 n +1} y^3 + (b x -n) y + c x^{1-n} \right ) w_y = 0 \]
Mathematica ✗
ClearAll[w, x, y, n, a, b, m, c, k]; pde = x*D[w[x, y], x] + (a*x^(2*n + 1)*y^3 + (b*x - n)*y + c*x^(1 - n))*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \text {Failed} \]
Maple ✓
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k'; pde := x*diff(w(x,y),x)+ (a*x^(2*n +1)*y^3 + (b*x-n)*y + c*x^(1-n) )*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( {b}^{3}\sum _{{\it \_R}=\RootOf \left ( {{\it \_Z}}^{3}a{c}^{2}+{\it \_Z}\,{b}^{3}-{b}^{3} \right ) }{\frac {1}{3\,{{\it \_R}}^{2}a{c}^{2}+{b}^{3}}\ln \left ( -{\frac {{x}^{n}by+{\it \_R}\,c}{c}} \right ) }-bx \right ) \] Solution contains RootOf
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Added January 2, 2019.
Problem 2.2.5.29 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ x w_x + \left ( a x^{n+2} y^3+ (b x^n-1) y + c x^{n-1} \right ) w_y = 0 \]
Mathematica ✗
ClearAll[w, x, y, n, a, b, m, c, k]; pde = x*D[w[x, y], x] + (a*x^(n + 2)*y^3 + (b*x^n - 1)*y + c*x^(n - 1))*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \text {Failed} \]
Maple ✓
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k'; pde := x*diff(w(x,y),x)+ (a*x^(n+2)*y^3+ (b*x^n-1)*y + c*x^(n-1) )*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1}{n} \left ( {b}^{3}\sum _{{\it \_R}=\RootOf \left ( {{\it \_Z}}^{3}a{c}^{2}+{\it \_Z}\,{b}^{3}-{b}^{3} \right ) }{\frac {1}{3\,{{\it \_R}}^{2}a{c}^{2}+{b}^{3}}\ln \left ( -{\frac {bxy+{\it \_R}\,c}{c}} \right ) }n-b{x}^{n} \right ) } \right ) \] Solution contains RootOf
____________________________________________________________________________________
Added January 2, 2019.
Problem 2.2.5.30 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ x w_x + \left ( y+a x^{n - m }y^m+b x^{n-k} y^k \right ) w_y = 0 \]
Mathematica ✗
ClearAll[w, x, y, n, a, b, m, c, k]; pde = x*D[w[x, y], x] + (y + a*x^(n - m)*y^m + b*x^(n - k)*y^k)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \text {Failed} \]
Maple ✓
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k'; pde := x*diff(w(x,y),x)+ ( y+a*x^(n - m)*y^m+b*x^(n-k)*y^k )*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1}{ \left ( n-1 \right ) x} \left ( \int _{{\it \_b}}^{y}\!-{\frac {{x}^{m}{x}^{k}}{x \left ( a{x}^{k}{{\it \_a}}^{m}+{{\it \_a}}^{k}{x}^{m}b \right ) }}\,{\rm d}{\it \_a}xn-\int _{{\it \_b}}^{y}\!-{\frac {{x}^{m}{x}^{k}}{x \left ( a{x}^{k}{{\it \_a}}^{m}+{{\it \_a}}^{k}{x}^{m}b \right ) }}\,{\rm d}{\it \_a}x+{x}^{n} \right ) } \right ) \]
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Added January 2, 2019.
Problem 2.2.5.31 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ y w_x + \left (x^{n-1}((1+2 n)x+a n) y-n x^{2 n}(x+a) \right ) w_y = 0 \]
Mathematica ✗
ClearAll[w, x, y, n, a, b, m, c, k]; pde = y*D[w[x, y], x] + (x^(n - 1)*((1 + 2*n)*x + a*n)*y - n*x^(2*n)*(x + a))*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \text {Failed} \]
Maple ✓
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k'; pde := y*diff(w(x,y),x)+ ( x^(n-1)*((1+2*n)*x+a*n)*y-n*x^(2*n)*(x+a) )*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( -1/2\,{\frac {1}{x} \left ( \sqrt {-{n}^{2}}\int ^{-2\,{\frac {1}{\sqrt {-{n}^{2}}}\arctan \left ( {\frac {n \left ( 2\,{x}^{n}a+{x}^{n+1}-y \right ) }{\sqrt {-{n}^{2}} \left ( {x}^{n+1}-y \right ) }} \right ) }}\!\tan \left ( 1/2\,{\it \_a}\,\sqrt {-{n}^{2}} \right ) {{\rm e}^{-{\it \_a}}}{d{\it \_a}}x-2\,{{\rm e}^{2\,{\frac {1}{\sqrt {-{n}^{2}}}\arctan \left ( {\frac {n \left ( 2\,{x}^{n}a+{x}^{n+1}-y \right ) }{\sqrt {-{n}^{2}} \left ( {x}^{n+1}-y \right ) }} \right ) }}}an-{{\rm e}^{2\,{\frac {1}{\sqrt {-{n}^{2}}}\arctan \left ( {\frac {n \left ( 2\,{x}^{n}a+{x}^{n+1}-y \right ) }{\sqrt {-{n}^{2}} \left ( {x}^{n+1}-y \right ) }} \right ) }}}nx \right ) } \right ) \]
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Added January 2, 2019.
Problem 2.2.5.32 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ y w_x + \left ( (a(2 n +k)x^k+b)x^{n-1}y -(a^2 n x^{2 k}+ a b x^k -c) x^{2 n-1} \right ) w_y = 0 \]
Mathematica ✗
ClearAll[w, x, y, n, a, b, m, c, k]; pde = y*D[w[x, y], x] + ((a*(2*n + k)*x^k + b)*x^(n - 1)*y - (a^2*n*x^(2*k) + a*b*x^k - c)*x^(2*n - 1))*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \text {Failed} \]
Maple ✗
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k'; pde := y*diff(w(x,y),x)+ ( (a*(2*n+k)*x^k+b)*x^(n-1)*y -(a^2*n*x^(2*k)+ a*b*x^k-c)*x^(2*n-1) )*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ \text { sol=() } \]
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Added January 2, 2019.
Problem 2.2.5.33 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ x(2 a x y+b) w_x - \left ( a(m+3) x y^2+b(m+2)y-c x^m \right ) w_y = 0 \]
Mathematica ✓
ClearAll[w, x, y, n, a, b, m, c, k]; pde = x*(2*a*x*y + b)*D[w[x, y], x] - (a*(m + 3)*x*y^2 + b*(m + 2)*y - c*x^m)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to c_1\left (-\frac {x^{m+2} \left (-2 a m x y^2-2 a x y^2-2 b m y-2 b y+c x^m\right )}{2 a (m+1)}\right )\right \}\right \} \]
Maple ✓
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k'; pde := x*(2*a*x*y+b)*diff(w(x,y),x)- ( a*(m+3)*x*y^2+b*(m+2)*y-c*x^m )*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( -1/2\,{\frac { \left ( 2\,ax{y}^{2}m+2\,ax{y}^{2}+2\,bym+2\,by-c{x}^{m} \right ) {x}^{2}{x}^{m}}{m+1}} \right ) \]
____________________________________________________________________________________
Added January 2, 2019.
Problem 2.2.5.34 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ x^2(2 a x y+b) w_x - \left ( 4 a x^2 y^2 + 3 b x y-c x^2 - k \right ) w_y = 0 \]
Mathematica ✓
ClearAll[w, x, y, n, a, b, m, c, k]; pde = x^2*(2*a*x*y + b)*D[w[x, y], x] - (4*a*x^2*y^2 + 3*b*x*y - c*x^2 - k)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to c_1\left (\frac {4 a x^4 y^2+4 b x^3 y-c x^4-2 k x^2}{4 a}\right )\right \}\right \} \]
Maple ✓
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k'; pde := x^2*(2*a*x*y+b)*diff(w(x,y),x)- ( 4*a*x^2*y^2 + 3*b*x*y-c*x^2 - k )*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( -a{x}^{4}{y}^{2}-b{x}^{3}y+1/4\,{x}^{4}c+1/2\,k{x}^{2} \right ) \]
____________________________________________________________________________________
Added January 2, 2019.
Problem 2.2.5.35 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a x^m w_x + b y^n w_y = 0 \]
Mathematica ✓
ClearAll[w, x, y, n, a, b, m, c, k]; pde = a*x^m*D[w[x, y], x] + b*y^n*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to c_1\left (-\frac {x^{-m} y^{-n} \left (-a y x^m+a m y x^m+b x y^n-b n x y^n\right )}{a (m-1) (n-1)}\right )\right \}\right \} \]
Maple ✓
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k'; pde :=a*x^m*diff(w(x,y),x)+ b*y^n*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac {{x}^{1-m}bn-{y}^{-n+1}am-{x}^{1-m}b+{y}^{-n+1}a}{a \left ( m-1 \right ) }} \right ) \]
____________________________________________________________________________________
Added January 2, 2019.
Problem 2.2.5.36 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a x^n w_x + (b y+ c x^m) w_y = 0 \]
Mathematica ✓
ClearAll[w, x, y, n, a, b, m, c, k]; pde = a*x^n*D[w[x, y], x] + (b*y + c*x^m)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to c_1\left (\frac {b^{-\frac {n}{n-1}} (a-a n)^{-\frac {m}{n-1}-\frac {1}{n-1}} e^{-\frac {b x^{1-n}}{a (1-n)}} \left (-c (a-a n)^{\frac {n}{n-1}} b^{\frac {m}{n-1}+\frac {1}{n-1}} e^{\frac {b x^{1-n}}{a (1-n)}} \text {Gamma}\left (\frac {-m+n-1}{n-1},\frac {b x^{1-n}}{a-a n}\right )-a y b^{\frac {n}{n-1}} (a-a n)^{\frac {m}{n-1}+\frac {1}{n-1}}+a n y b^{\frac {n}{n-1}} (a-a n)^{\frac {m}{n-1}+\frac {1}{n-1}}\right )}{a (n-1)}\right )\right \}\right \} \]
Maple ✓
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k'; pde :=a*x^n*diff(w(x,y),x)+ (b*y+c*x^m)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1}{ab \left ( {m}^{3}-6\,{m}^{2}n+11\,m{n}^{2}-6\,{n}^{3}+6\,{m}^{2}-22\,mn+18\,{n}^{2}+11\,m-18\,n+6 \right ) } \left ( -{{\rm e}^{1/2\,{\frac {{x}^{-n+1}b}{a \left ( n-1 \right ) }}}}{x}^{m} \left ( -{\frac {{x}^{-n+1}b}{a \left ( n-1 \right ) }} \right ) ^{1/2\,{\frac {m-2\,n+2}{n-1}}} \WhittakerM \left ( -1/2\,{\frac {m-2\,n+2}{n-1}},-1/2\,{\frac {m-3\,n+3}{n-1}},-{\frac {{x}^{-n+1}b}{a \left ( n-1 \right ) }} \right ) ac{m}^{2}+2\,{{\rm e}^{1/2\,{\frac {{x}^{-n+1}b}{a \left ( n-1 \right ) }}}}{x}^{-n+m+1} \left ( -{\frac {{x}^{-n+1}b}{a \left ( n-1 \right ) }} \right ) ^{1/2\,{\frac {m-2\,n+2}{n-1}}} \WhittakerM \left ( -1/2\,{\frac {m}{n-1}},-1/2\,{\frac {m-3\,n+3}{n-1}},-{\frac {{x}^{-n+1}b}{a \left ( n-1 \right ) }} \right ) bcn-4\,{{\rm e}^{1/2\,{\frac {{x}^{-n+1}b}{a \left ( n-1 \right ) }}}}{x}^{m} \left ( -{\frac {{x}^{-n+1}b}{a \left ( n-1 \right ) }} \right ) ^{1/2\,{\frac {m-2\,n+2}{n-1}}} \WhittakerM \left ( -1/2\,{\frac {m-2\,n+2}{n-1}},-1/2\,{\frac {m-3\,n+3}{n-1}},-{\frac {{x}^{-n+1}b}{a \left ( n-1 \right ) }} \right ) acm+12\,{{\rm e}^{1/2\,{\frac {{x}^{-n+1}b}{a \left ( n-1 \right ) }}}}{x}^{m} \left ( -{\frac {{x}^{-n+1}b}{a \left ( n-1 \right ) }} \right ) ^{1/2\,{\frac {m-2\,n+2}{n-1}}} \WhittakerM \left ( -1/2\,{\frac {m-2\,n+2}{n-1}},-1/2\,{\frac {m-3\,n+3}{n-1}},-{\frac {{x}^{-n+1}b}{a \left ( n-1 \right ) }} \right ) acn-12\,{{\rm e}^{1/2\,{\frac {{x}^{-n+1}b}{a \left ( n-1 \right ) }}}}{x}^{m} \left ( -{\frac {{x}^{-n+1}b}{a \left ( n-1 \right ) }} \right ) ^{1/2\,{\frac {m-2\,n+2}{n-1}}} \WhittakerM \left ( -1/2\,{\frac {m-2\,n+2}{n-1}},-1/2\,{\frac {m-3\,n+3}{n-1}},-{\frac {{x}^{-n+1}b}{a \left ( n-1 \right ) }} \right ) ac{n}^{2}+4\,{{\rm e}^{1/2\,{\frac {{x}^{-n+1}b}{a \left ( n-1 \right ) }}}}{x}^{m} \left ( -{\frac {{x}^{-n+1}b}{a \left ( n-1 \right ) }} \right ) ^{1/2\,{\frac {m-2\,n+2}{n-1}}} \WhittakerM \left ( -1/2\,{\frac {m-2\,n+2}{n-1}},-1/2\,{\frac {m-3\,n+3}{n-1}},-{\frac {{x}^{-n+1}b}{a \left ( n-1 \right ) }} \right ) ac{n}^{3}-6\,{{\rm e}^{1/2\,{\frac {{x}^{-n+1}b}{a \left ( n-1 \right ) }}}}{x}^{m} \left ( -{\frac {{x}^{-n+1}b}{a \left ( n-1 \right ) }} \right ) ^{1/2\,{\frac {m-2\,n+2}{n-1}}} \WhittakerM \left ( -1/2\,{\frac {m}{n-1}},-1/2\,{\frac {m-3\,n+3}{n-1}},-{\frac {{x}^{-n+1}b}{a \left ( n-1 \right ) }} \right ) ac{n}^{2}+6\,{{\rm e}^{1/2\,{\frac {{x}^{-n+1}b}{a \left ( n-1 \right ) }}}}{x}^{m} \left ( -{\frac {{x}^{-n+1}b}{a \left ( n-1 \right ) }} \right ) ^{1/2\,{\frac {m-2\,n+2}{n-1}}} \WhittakerM \left ( -1/2\,{\frac {m}{n-1}},-1/2\,{\frac {m-3\,n+3}{n-1}},-{\frac {{x}^{-n+1}b}{a \left ( n-1 \right ) }} \right ) acn-{{\rm e}^{1/2\,{\frac {{x}^{-n+1}b}{a \left ( n-1 \right ) }}}}{x}^{m} \left ( -{\frac {{x}^{-n+1}b}{a \left ( n-1 \right ) }} \right ) ^{1/2\,{\frac {m-2\,n+2}{n-1}}} \WhittakerM \left ( -1/2\,{\frac {m}{n-1}},-1/2\,{\frac {m-3\,n+3}{n-1}},-{\frac {{x}^{-n+1}b}{a \left ( n-1 \right ) }} \right ) acm+2\,{{\rm e}^{1/2\,{\frac {{x}^{-n+1}b}{a \left ( n-1 \right ) }}}}{x}^{m} \left ( -{\frac {{x}^{-n+1}b}{a \left ( n-1 \right ) }} \right ) ^{1/2\,{\frac {m-2\,n+2}{n-1}}} \WhittakerM \left ( -1/2\,{\frac {m}{n-1}},-1/2\,{\frac {m-3\,n+3}{n-1}},-{\frac {{x}^{-n+1}b}{a \left ( n-1 \right ) }} \right ) ac{n}^{3}-{{\rm e}^{1/2\,{\frac {{x}^{-n+1}b}{a \left ( n-1 \right ) }}}}{x}^{-n+m+1} \left ( -{\frac {{x}^{-n+1}b}{a \left ( n-1 \right ) }} \right ) ^{1/2\,{\frac {m-2\,n+2}{n-1}}} \WhittakerM \left ( -1/2\,{\frac {m}{n-1}},-1/2\,{\frac {m-3\,n+3}{n-1}},-{\frac {{x}^{-n+1}b}{a \left ( n-1 \right ) }} \right ) bc{n}^{2}-{{\rm e}^{1/2\,{\frac {{x}^{-n+1}b}{a \left ( n-1 \right ) }}}}{x}^{m} \left ( -{\frac {{x}^{-n+1}b}{a \left ( n-1 \right ) }} \right ) ^{1/2\,{\frac {m-2\,n+2}{n-1}}} \WhittakerM \left ( -1/2\,{\frac {m}{n-1}},-1/2\,{\frac {m-3\,n+3}{n-1}},-{\frac {{x}^{-n+1}b}{a \left ( n-1 \right ) }} \right ) acm{n}^{2}+{{\rm e}^{1/2\,{\frac {{x}^{-n+1}b}{a \left ( n-1 \right ) }}}}{x}^{m} \left ( -{\frac {{x}^{-n+1}b}{a \left ( n-1 \right ) }} \right ) ^{1/2\,{\frac {m-2\,n+2}{n-1}}} \WhittakerM \left ( -1/2\,{\frac {m-2\,n+2}{n-1}},-1/2\,{\frac {m-3\,n+3}{n-1}},-{\frac {{x}^{-n+1}b}{a \left ( n-1 \right ) }} \right ) ac{m}^{2}n+2\,{{\rm e}^{1/2\,{\frac {{x}^{-n+1}b}{a \left ( n-1 \right ) }}}}{x}^{m} \left ( -{\frac {{x}^{-n+1}b}{a \left ( n-1 \right ) }} \right ) ^{1/2\,{\frac {m-2\,n+2}{n-1}}} \WhittakerM \left ( -1/2\,{\frac {m}{n-1}},-1/2\,{\frac {m-3\,n+3}{n-1}},-{\frac {{x}^{-n+1}b}{a \left ( n-1 \right ) }} \right ) acmn-4\,{{\rm e}^{1/2\,{\frac {{x}^{-n+1}b}{a \left ( n-1 \right ) }}}}{x}^{m} \left ( -{\frac {{x}^{-n+1}b}{a \left ( n-1 \right ) }} \right ) ^{1/2\,{\frac {m-2\,n+2}{n-1}}} \WhittakerM \left ( -1/2\,{\frac {m-2\,n+2}{n-1}},-1/2\,{\frac {m-3\,n+3}{n-1}},-{\frac {{x}^{-n+1}b}{a \left ( n-1 \right ) }} \right ) acm{n}^{2}+8\,{{\rm e}^{1/2\,{\frac {{x}^{-n+1}b}{a \left ( n-1 \right ) }}}}{x}^{m} \left ( -{\frac {{x}^{-n+1}b}{a \left ( n-1 \right ) }} \right ) ^{1/2\,{\frac {m-2\,n+2}{n-1}}} \WhittakerM \left ( -1/2\,{\frac {m-2\,n+2}{n-1}},-1/2\,{\frac {m-3\,n+3}{n-1}},-{\frac {{x}^{-n+1}b}{a \left ( n-1 \right ) }} \right ) acmn-6\,{{\rm e}^{{\frac {{x}^{-n+1}b}{a \left ( n-1 \right ) }}}}yab{m}^{2}n+11\,{{\rm e}^{{\frac {{x}^{-n+1}b}{a \left ( n-1 \right ) }}}}yabm{n}^{2}-22\,{{\rm e}^{{\frac {{x}^{-n+1}b}{a \left ( n-1 \right ) }}}}yabmn+6\,ya{{\rm e}^{{\frac {{x}^{-n+1}b}{a \left ( n-1 \right ) }}}}b+11\,{{\rm e}^{{\frac {{x}^{-n+1}b}{a \left ( n-1 \right ) }}}}yabm-18\,{{\rm e}^{{\frac {{x}^{-n+1}b}{a \left ( n-1 \right ) }}}}yabn+{{\rm e}^{{\frac {{x}^{-n+1}b}{a \left ( n-1 \right ) }}}}yab{m}^{3}-6\,{{\rm e}^{{\frac {{x}^{-n+1}b}{a \left ( n-1 \right ) }}}}yab{n}^{3}+6\,{{\rm e}^{{\frac {{x}^{-n+1}b}{a \left ( n-1 \right ) }}}}yab{m}^{2}+18\,{{\rm e}^{{\frac {{x}^{-n+1}b}{a \left ( n-1 \right ) }}}}yab{n}^{2}-4\,{{\rm e}^{1/2\,{\frac {{x}^{-n+1}b}{a \left ( n-1 \right ) }}}}{x}^{m} \left ( -{\frac {{x}^{-n+1}b}{a \left ( n-1 \right ) }} \right ) ^{1/2\,{\frac {m-2\,n+2}{n-1}}} \WhittakerM \left ( -1/2\,{\frac {m-2\,n+2}{n-1}},-1/2\,{\frac {m-3\,n+3}{n-1}},-{\frac {{x}^{-n+1}b}{a \left ( n-1 \right ) }} \right ) ac-{{\rm e}^{1/2\,{\frac {{x}^{-n+1}b}{a \left ( n-1 \right ) }}}}{x}^{-n+m+1} \left ( -{\frac {{x}^{-n+1}b}{a \left ( n-1 \right ) }} \right ) ^{1/2\,{\frac {m-2\,n+2}{n-1}}} \WhittakerM \left ( -1/2\,{\frac {m}{n-1}},-1/2\,{\frac {m-3\,n+3}{n-1}},-{\frac {{x}^{-n+1}b}{a \left ( n-1 \right ) }} \right ) bc-2\,{{\rm e}^{1/2\,{\frac {{x}^{-n+1}b}{a \left ( n-1 \right ) }}}}{x}^{m} \left ( -{\frac {{x}^{-n+1}b}{a \left ( n-1 \right ) }} \right ) ^{1/2\,{\frac {m-2\,n+2}{n-1}}} \WhittakerM \left ( -1/2\,{\frac {m}{n-1}},-1/2\,{\frac {m-3\,n+3}{n-1}},-{\frac {{x}^{-n+1}b}{a \left ( n-1 \right ) }} \right ) ac \right ) } \right ) \]
____________________________________________________________________________________
Added January 2, 2019.
Problem 2.2.5.37 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a x^k w_x + (y^n+ b x^m y) w_y = 0 \]
Mathematica ✗
ClearAll[w, x, y, n, a, b, m, c, k]; pde = a*x^k*D[w[x, y], x] + (y^n + b*x^m*y)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}, Assumptions -> {n != 1}], 60*10]];
\[ \text {\$Aborted} \] Timed out
Maple ✓
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k'; pde :=a*x^k*diff(w(x,y),x)+ (y^n+b*x^m*y)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) assuming n<>1),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1}{a} \left ( {\frac {a}{{y}^{ \left ( k-m-1 \right ) ^{-1}}}{y}^{{\frac {mn}{k-m-1}}}{y}^{{\frac {n}{k-m-1}}}{y}^{{\frac {k}{k-m-1}}}{{\rm e}^{{\frac {b{x}^{-k+m+1}}{ \left ( k-m-1 \right ) a}}}} \left ( {y}^{{\frac {kn}{k-m-1}}} \right ) ^{-1} \left ( {{\rm e}^{{\frac {b{x}^{-k+m+1}n}{ \left ( k-m-1 \right ) a}}}} \right ) ^{-1} \left ( {y}^{{\frac {m}{k-m-1}}} \right ) ^{-1}}+{\frac {n}{-k+m+1} \left ( {\frac {b \left ( n-1 \right ) }{ \left ( k-m-1 \right ) a}} \right ) ^{{\frac {k}{-k+m+1}}- \left ( -k+m+1 \right ) ^{-1}} \left ( {\frac { \left ( k-m-1 \right ) ^{2}a}{ \left ( k-1 \right ) \left ( 2\,k-2-m \right ) \left ( 3\,k-3-2\,m \right ) b \left ( n-1 \right ) }{x}^{{\frac {{k}^{2}}{-k+m+1}}-{\frac {km}{-k+m+1}}-2\,{\frac {k}{-k+m+1}}+{\frac {m}{-k+m+1}}+ \left ( -k+m+1 \right ) ^{-1}+k-m-1} \left ( {\frac {b \left ( n-1 \right ) }{ \left ( k-m-1 \right ) a}} \right ) ^{-{\frac {k}{-k+m+1}}+ \left ( -k+m+1 \right ) ^{-1}} \left ( {\frac {b{x}^{-k+m+1} \left ( n-1 \right ) {k}^{2}}{ \left ( k-m-1 \right ) a}}-2\,{\frac {b{x}^{-k+m+1} \left ( n-1 \right ) km}{ \left ( k-m-1 \right ) a}}+{\frac {b{x}^{-k+m+1} \left ( n-1 \right ) {m}^{2}}{ \left ( k-m-1 \right ) a}}-2\,{\frac {b{x}^{-k+m+1} \left ( n-1 \right ) k}{ \left ( k-m-1 \right ) a}}+2\,{\frac {b{x}^{-k+m+1} \left ( n-1 \right ) m}{ \left ( k-m-1 \right ) a}}+2\,{k}^{2}-3\,km+{m}^{2}+{\frac {b{x}^{-k+m+1} \left ( n-1 \right ) }{ \left ( k-m-1 \right ) a}}-4\,k+3\,m+2 \right ) \left ( {\frac {b{x}^{-k+m+1} \left ( n-1 \right ) }{ \left ( k-m-1 \right ) a}} \right ) ^{-1/2\,{\frac {2\,k-2-m}{k-m-1}}}{{\rm e}^{-1/2\,{\frac {b{x}^{-k+m+1} \left ( n-1 \right ) }{ \left ( k-m-1 \right ) a}}}} \WhittakerM \left ( {\frac {k-1}{k-m-1}}-1/2\,{\frac {2\,k-2-m}{k-m-1}},1/2\,{\frac {2\,k-2-m}{k-m-1}}+1/2,{\frac {b{x}^{-k+m+1} \left ( n-1 \right ) }{ \left ( k-m-1 \right ) a}} \right ) }+{\frac { \left ( k-m-1 \right ) ^{2} \left ( 2\,k-2-m \right ) a}{ \left ( k-1 \right ) \left ( 3\,k-3-2\,m \right ) b \left ( n-1 \right ) }{x}^{{\frac {{k}^{2}}{-k+m+1}}-{\frac {km}{-k+m+1}}-2\,{\frac {k}{-k+m+1}}+{\frac {m}{-k+m+1}}+ \left ( -k+m+1 \right ) ^{-1}+k-m-1} \left ( {\frac {b \left ( n-1 \right ) }{ \left ( k-m-1 \right ) a}} \right ) ^{-{\frac {k}{-k+m+1}}+ \left ( -k+m+1 \right ) ^{-1}} \left ( {\frac {b{x}^{-k+m+1} \left ( n-1 \right ) }{ \left ( k-m-1 \right ) a}} \right ) ^{-1/2\,{\frac {2\,k-2-m}{k-m-1}}}{{\rm e}^{-1/2\,{\frac {b{x}^{-k+m+1} \left ( n-1 \right ) }{ \left ( k-m-1 \right ) a}}}} \WhittakerM \left ( {\frac {k-1}{k-m-1}}-1/2\,{\frac {2\,k-2-m}{k-m-1}}+1,1/2\,{\frac {2\,k-2-m}{k-m-1}}+1/2,{\frac {b{x}^{-k+m+1} \left ( n-1 \right ) }{ \left ( k-m-1 \right ) a}} \right ) } \right ) }-{\frac {1}{-k+m+1} \left ( {\frac {b \left ( n-1 \right ) }{ \left ( k-m-1 \right ) a}} \right ) ^{{\frac {k}{-k+m+1}}- \left ( -k+m+1 \right ) ^{-1}} \left ( {\frac { \left ( k-m-1 \right ) ^{2}a}{ \left ( k-1 \right ) \left ( 2\,k-2-m \right ) \left ( 3\,k-3-2\,m \right ) b \left ( n-1 \right ) }{x}^{{\frac {{k}^{2}}{-k+m+1}}-{\frac {km}{-k+m+1}}-2\,{\frac {k}{-k+m+1}}+{\frac {m}{-k+m+1}}+ \left ( -k+m+1 \right ) ^{-1}+k-m-1} \left ( {\frac {b \left ( n-1 \right ) }{ \left ( k-m-1 \right ) a}} \right ) ^{-{\frac {k}{-k+m+1}}+ \left ( -k+m+1 \right ) ^{-1}} \left ( {\frac {b{x}^{-k+m+1} \left ( n-1 \right ) {k}^{2}}{ \left ( k-m-1 \right ) a}}-2\,{\frac {b{x}^{-k+m+1} \left ( n-1 \right ) km}{ \left ( k-m-1 \right ) a}}+{\frac {b{x}^{-k+m+1} \left ( n-1 \right ) {m}^{2}}{ \left ( k-m-1 \right ) a}}-2\,{\frac {b{x}^{-k+m+1} \left ( n-1 \right ) k}{ \left ( k-m-1 \right ) a}}+2\,{\frac {b{x}^{-k+m+1} \left ( n-1 \right ) m}{ \left ( k-m-1 \right ) a}}+2\,{k}^{2}-3\,km+{m}^{2}+{\frac {b{x}^{-k+m+1} \left ( n-1 \right ) }{ \left ( k-m-1 \right ) a}}-4\,k+3\,m+2 \right ) \left ( {\frac {b{x}^{-k+m+1} \left ( n-1 \right ) }{ \left ( k-m-1 \right ) a}} \right ) ^{-1/2\,{\frac {2\,k-2-m}{k-m-1}}}{{\rm e}^{-1/2\,{\frac {b{x}^{-k+m+1} \left ( n-1 \right ) }{ \left ( k-m-1 \right ) a}}}} \WhittakerM \left ( {\frac {k-1}{k-m-1}}-1/2\,{\frac {2\,k-2-m}{k-m-1}},1/2\,{\frac {2\,k-2-m}{k-m-1}}+1/2,{\frac {b{x}^{-k+m+1} \left ( n-1 \right ) }{ \left ( k-m-1 \right ) a}} \right ) }+{\frac { \left ( k-m-1 \right ) ^{2} \left ( 2\,k-2-m \right ) a}{ \left ( k-1 \right ) \left ( 3\,k-3-2\,m \right ) b \left ( n-1 \right ) }{x}^{{\frac {{k}^{2}}{-k+m+1}}-{\frac {km}{-k+m+1}}-2\,{\frac {k}{-k+m+1}}+{\frac {m}{-k+m+1}}+ \left ( -k+m+1 \right ) ^{-1}+k-m-1} \left ( {\frac {b \left ( n-1 \right ) }{ \left ( k-m-1 \right ) a}} \right ) ^{-{\frac {k}{-k+m+1}}+ \left ( -k+m+1 \right ) ^{-1}} \left ( {\frac {b{x}^{-k+m+1} \left ( n-1 \right ) }{ \left ( k-m-1 \right ) a}} \right ) ^{-1/2\,{\frac {2\,k-2-m}{k-m-1}}}{{\rm e}^{-1/2\,{\frac {b{x}^{-k+m+1} \left ( n-1 \right ) }{ \left ( k-m-1 \right ) a}}}} \WhittakerM \left ( {\frac {k-1}{k-m-1}}-1/2\,{\frac {2\,k-2-m}{k-m-1}}+1,1/2\,{\frac {2\,k-2-m}{k-m-1}}+1/2,{\frac {b{x}^{-k+m+1} \left ( n-1 \right ) }{ \left ( k-m-1 \right ) a}} \right ) } \right ) } \right ) } \right ) \]
____________________________________________________________________________________
Added January 2, 2019.
Problem 2.2.5.38 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ x(a x^k+b) w_x + \left ( \alpha x^n y^2+(\beta -a n x^k)y+\gamma x^{-n} \right ) w_y = 0 \]
Mathematica ✓
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma]; pde = x*(a*x^k + b)*D[w[x, y], x] + (alpha*x^n*y^2 + (beta - a*n*x^k)*y + gamma/x^n)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to c_1\left (\frac {-\beta \exp \left (\frac {\sqrt {\alpha } \sqrt {\gamma } \left (k \log (x)-\log \left (a x^k+b\right )\right ) \left (\sqrt {\frac {(-b n-\beta )^2}{\alpha \gamma }-4}-\frac {-b n-\beta }{\sqrt {\alpha } \sqrt {\gamma }}\right )}{2 b k}\right )-b n \exp \left (\frac {\sqrt {\alpha } \sqrt {\gamma } \left (k \log (x)-\log \left (a x^k+b\right )\right ) \left (\sqrt {\frac {(-b n-\beta )^2}{\alpha \gamma }-4}-\frac {-b n-\beta }{\sqrt {\alpha } \sqrt {\gamma }}\right )}{2 b k}\right )-\sqrt {\alpha } \sqrt {\gamma } \sqrt {\frac {(-b n-\beta )^2}{\alpha \gamma }-4} \exp \left (\frac {\sqrt {\alpha } \sqrt {\gamma } \left (k \log (x)-\log \left (a x^k+b\right )\right ) \left (\sqrt {\frac {(-b n-\beta )^2}{\alpha \gamma }-4}-\frac {-b n-\beta }{\sqrt {\alpha } \sqrt {\gamma }}\right )}{2 b k}\right )-2 \alpha y x^n \exp \left (\frac {\sqrt {\alpha } \sqrt {\gamma } \left (k \log (x)-\log \left (a x^k+b\right )\right ) \left (\sqrt {\frac {(-b n-\beta )^2}{\alpha \gamma }-4}-\frac {-b n-\beta }{\sqrt {\alpha } \sqrt {\gamma }}\right )}{2 b k}\right )}{\beta \exp \left (\frac {\sqrt {\alpha } \sqrt {\gamma } \left (k \log (x)-\log \left (a x^k+b\right )\right ) \left (-\frac {-b n-\beta }{\sqrt {\alpha } \sqrt {\gamma }}-\sqrt {\frac {(-b n-\beta )^2}{\alpha \gamma }-4}\right )}{2 b k}\right )+b n \exp \left (\frac {\sqrt {\alpha } \sqrt {\gamma } \left (k \log (x)-\log \left (a x^k+b\right )\right ) \left (-\frac {-b n-\beta }{\sqrt {\alpha } \sqrt {\gamma }}-\sqrt {\frac {(-b n-\beta )^2}{\alpha \gamma }-4}\right )}{2 b k}\right )-\sqrt {\alpha } \sqrt {\gamma } \sqrt {\frac {(-b n-\beta )^2}{\alpha \gamma }-4} \exp \left (\frac {\sqrt {\alpha } \sqrt {\gamma } \left (k \log (x)-\log \left (a x^k+b\right )\right ) \left (-\frac {-b n-\beta }{\sqrt {\alpha } \sqrt {\gamma }}-\sqrt {\frac {(-b n-\beta )^2}{\alpha \gamma }-4}\right )}{2 b k}\right )+2 \alpha y x^n \exp \left (\frac {\sqrt {\alpha } \sqrt {\gamma } \left (k \log (x)-\log \left (a x^k+b\right )\right ) \left (-\frac {-b n-\beta }{\sqrt {\alpha } \sqrt {\gamma }}-\sqrt {\frac {(-b n-\beta )^2}{\alpha \gamma }-4}\right )}{2 b k}\right )}\right )\right \}\right \} \]
Maple ✓
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g'; pde :=x*(a*x^k+b)*diff(w(x,y),x)+ (alpha*x^n*y^2+(beta-a*n*x^k)*y+g*x^(-n))*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1}{\sqrt {- \left ( bn+\beta \right ) ^{2} \left ( -{b}^{2}{n}^{2}-2\,b\beta \,n+4\,\alpha \,g-{\beta }^{2} \right ) }bk} \left ( -2\,{b}^{3}k{n}^{2}\arctanh \left ( {\frac {2\,{x}^{n}y\alpha \,bn+2\,{x}^{n}y\alpha \,\beta +{b}^{2}{n}^{2}+2\,b\beta \,n+{\beta }^{2}}{\sqrt {- \left ( bn+\beta \right ) ^{2} \left ( -{b}^{2}{n}^{2}-2\,b\beta \,n+4\,\alpha \,g-{\beta }^{2} \right ) }}} \right ) -4\,{b}^{2}\beta \,kn\arctanh \left ( {\frac {2\,{x}^{n}y\alpha \,bn+2\,{x}^{n}y\alpha \,\beta +{b}^{2}{n}^{2}+2\,b\beta \,n+{\beta }^{2}}{\sqrt {- \left ( bn+\beta \right ) ^{2} \left ( -{b}^{2}{n}^{2}-2\,b\beta \,n+4\,\alpha \,g-{\beta }^{2} \right ) }}} \right ) -\sqrt {- \left ( bn+\beta \right ) ^{2} \left ( -{b}^{2}{n}^{2}-2\,b\beta \,n+4\,\alpha \,g-{\beta }^{2} \right ) }\ln \left ( x \right ) bkn-2\,b{\beta }^{2}k\arctanh \left ( {\frac {2\,{x}^{n}y\alpha \,bn+2\,{x}^{n}y\alpha \,\beta +{b}^{2}{n}^{2}+2\,b\beta \,n+{\beta }^{2}}{\sqrt {- \left ( bn+\beta \right ) ^{2} \left ( -{b}^{2}{n}^{2}-2\,b\beta \,n+4\,\alpha \,g-{\beta }^{2} \right ) }}} \right ) +\sqrt {- \left ( bn+\beta \right ) ^{2} \left ( -{b}^{2}{n}^{2}-2\,b\beta \,n+4\,\alpha \,g-{\beta }^{2} \right ) }\ln \left ( {x}^{k}a+b \right ) bn-\sqrt {- \left ( bn+\beta \right ) ^{2} \left ( -{b}^{2}{n}^{2}-2\,b\beta \,n+4\,\alpha \,g-{\beta }^{2} \right ) }\ln \left ( x \right ) \beta \,k+\sqrt {- \left ( bn+\beta \right ) ^{2} \left ( -{b}^{2}{n}^{2}-2\,b\beta \,n+4\,\alpha \,g-{\beta }^{2} \right ) }\ln \left ( {x}^{k}a+b \right ) \beta \right ) } \right ) \]
____________________________________________________________________________________
Added January 2, 2019.
Problem 2.2.5.39 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ (y+ A x^n + a) w_x - \left ( n A x^{n-1} y + k x^m + b\right ) w_y = 0 \]
Mathematica ✓
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A]; pde = (y + A*x^n + a)*D[w[x, y], x] - (n*A*x^(n - 1)*y + k*x^m + b)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to c_1\left (\frac {2 a m y+2 a y+2 A m y x^n+2 A y x^n+2 b m x+2 b x+2 k x^{m+1}+m y^2+y^2}{m+1}\right )\right \}\right \} \]
Maple ✓
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A'; pde :=(y+ A*x^n + a)*diff(w(x,y),x)- ( n*A*x^(n-1)*y + k*x^m + b)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( -1/2\,{\frac {2\,A{x}^{n}ym+2\,A{x}^{n}y+{y}^{2}m+2\,aym+2\,k{x}^{m}x+2\,bxm+{y}^{2}+2\,ya+2\,bx}{m+1}} \right ) \]
____________________________________________________________________________________
Added January 2, 2019.
Problem 2.2.5.40 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ (y+ a x^{n+1}+b x^n) w_x + \left (a n x^n + c x^{n-1} \right ) y w_y = 0 \]
Mathematica ✗
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A]; pde = (y + a*x^(n + 1) + b*x^n)*D[w[x, y], x] + (a*n*x^n + c*x^(n - 1))*y*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \text {Failed} \]
Maple ✗
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A'; pde :=(y+ a*x^(n+1)+b*x^n)*diff(w(x,y),x)+ ( a*n*x^n + c*x^(n-1))*y*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ \text { sol=() } \]
____________________________________________________________________________________
Added January 2, 2019.
Problem 2.2.5.41 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ x(2 a x^n y+b) w_x - \left (a(3 n+m)x^n y^2+b(2 n+m)y-A x^m -C x^{-n} \right ) w_y = 0 \]
Mathematica ✓
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0]; pde = x*(2*a*x^n*y + b)*D[w[x, y], x] - (a*(3*n + m)*x^n*y^2 + b*(2*n + m)*y - A*x^m - C0/x^n)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to c_1\left (\frac {x^{m+n} \left (2 a m y^2 x^{2 n}+2 a n y^2 x^{2 n}-A x^{m+n}+2 b m y x^n+2 b n y x^n-2 \text {C0}\right )}{2 a (m+n)}\right )\right \}\right \} \]
Maple ✓
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A'; C:='C'; pde :=x*(2*a*x^n*y+b)*diff(w(x,y),x)- ( a*(3*n+m)*x^n*y^2+b*(2*n+m)*y-A*x^m -C*x^(-n))*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( 1/2\,{\frac {-2\,{x}^{3\,n+m}{y}^{2}am-2\,{x}^{3\,n+m}{y}^{2}an-2\,{x}^{m+2\,n}ybm-2\,{x}^{m+2\,n}ybn+{x}^{2\,n+2\,m}A+2\,{x}^{n+m}C}{n+m}} \right ) \]
____________________________________________________________________________________
Added January 2, 2019.
Problem 2.2.5.42 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ (a x^n+b x^2+ x y) w_x + \left (c x^n + b x y+ y^2 \right ) w_y = 0 \]
Mathematica ✗
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0]; pde = (a*x^n + b*x^2 + x*y)*D[w[x, y], x] + (c*x^n + b*x*y + y^2)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \text {Failed} \]
Maple ✓
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A'; C:='C'; pde :=(a*x^n+b*x^2+ x*y)*diff(w(x,y),x)+ ( c*x^n + b*x*y+ y^2)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( 1/3\,{\frac {1}{n-2} \left ( \ln \left ( ab{x}^{2}+{a}^{2}{x}^{n}+c{x}^{2} \right ) {n}^{2}-2\,\ln \left ( x \right ) {n}^{2}+{\frac {n}{ \left ( n-2 \right ) \left ( n-1 \right ) } \left ( -{n}^{3}\ln \left ( 9\,{\frac {ab{x}^{2}{n}^{2}+{x}^{n}{a}^{2}{n}^{2}-3\,nab{x}^{2}+c{x}^{2}{n}^{2}-3\,{a}^{2}{x}^{n}n+3\,ab{x}^{2}-3\,cn{x}^{2}+3\,{a}^{2}{x}^{n}+3\,c{x}^{2}}{ \left ( {x}^{2}nb+{x}^{n}an+xny-3\,b{x}^{2}-3\,{x}^{n}a-3\,yx \right ) a}} \right ) +\ln \left ( -9\,{\frac { \left ( a{n}^{2}y-c{n}^{2}x-3\,any+3\,cnx+3\,ya-3\,cx \right ) x}{ \left ( {x}^{n}a+b{x}^{2}+yx \right ) an}} \right ) {n}^{3}+{n}^{2}\ln \left ( 9\,{\frac {{x}^{2}{n}^{3}ba+{x}^{n}{n}^{3}{a}^{2}+x{n}^{3}ya-4\,ab{x}^{2}{n}^{2}-4\,{x}^{n}{a}^{2}{n}^{2}-5\,a{n}^{2}xy+6\,nab{x}^{2}+c{x}^{2}{n}^{2}+6\,{a}^{2}{x}^{n}n+9\,axny-3\,ab{x}^{2}-3\,cn{x}^{2}-3\,{a}^{2}{x}^{n}-6\,axy+3\,c{x}^{2}}{ \left ( 2\,{x}^{2}nb+2\,{x}^{n}an+2\,xny-3\,b{x}^{2}-3\,{x}^{n}a-3\,yx \right ) a}} \right ) +4\,{n}^{2}\ln \left ( 9\,{\frac {ab{x}^{2}{n}^{2}+{x}^{n}{a}^{2}{n}^{2}-3\,nab{x}^{2}+c{x}^{2}{n}^{2}-3\,{a}^{2}{x}^{n}n+3\,ab{x}^{2}-3\,cn{x}^{2}+3\,{a}^{2}{x}^{n}+3\,c{x}^{2}}{ \left ( {x}^{2}nb+{x}^{n}an+xny-3\,b{x}^{2}-3\,{x}^{n}a-3\,yx \right ) a}} \right ) -5\,\ln \left ( -9\,{\frac { \left ( a{n}^{2}y-c{n}^{2}x-3\,any+3\,cnx+3\,ya-3\,cx \right ) x}{ \left ( {x}^{n}a+b{x}^{2}+yx \right ) an}} \right ) {n}^{2}-3\,n\ln \left ( 9\,{\frac {{x}^{2}{n}^{3}ba+{x}^{n}{n}^{3}{a}^{2}+x{n}^{3}ya-4\,ab{x}^{2}{n}^{2}-4\,{x}^{n}{a}^{2}{n}^{2}-5\,a{n}^{2}xy+6\,nab{x}^{2}+c{x}^{2}{n}^{2}+6\,{a}^{2}{x}^{n}n+9\,axny-3\,ab{x}^{2}-3\,cn{x}^{2}-3\,{a}^{2}{x}^{n}-6\,axy+3\,c{x}^{2}}{ \left ( 2\,{x}^{2}nb+2\,{x}^{n}an+2\,xny-3\,b{x}^{2}-3\,{x}^{n}a-3\,yx \right ) a}} \right ) -6\,n\ln \left ( 9\,{\frac {ab{x}^{2}{n}^{2}+{x}^{n}{a}^{2}{n}^{2}-3\,nab{x}^{2}+c{x}^{2}{n}^{2}-3\,{a}^{2}{x}^{n}n+3\,ab{x}^{2}-3\,cn{x}^{2}+3\,{a}^{2}{x}^{n}+3\,c{x}^{2}}{ \left ( {x}^{2}nb+{x}^{n}an+xny-3\,b{x}^{2}-3\,{x}^{n}a-3\,yx \right ) a}} \right ) +9\,\ln \left ( -9\,{\frac { \left ( a{n}^{2}y-c{n}^{2}x-3\,any+3\,cnx+3\,ya-3\,cx \right ) x}{ \left ( {x}^{n}a+b{x}^{2}+yx \right ) an}} \right ) n+3\,\ln \left ( 9\,{\frac {{x}^{2}{n}^{3}ba+{x}^{n}{n}^{3}{a}^{2}+x{n}^{3}ya-4\,ab{x}^{2}{n}^{2}-4\,{x}^{n}{a}^{2}{n}^{2}-5\,a{n}^{2}xy+6\,nab{x}^{2}+c{x}^{2}{n}^{2}+6\,{a}^{2}{x}^{n}n+9\,axny-3\,ab{x}^{2}-3\,cn{x}^{2}-3\,{a}^{2}{x}^{n}-6\,axy+3\,c{x}^{2}}{ \left ( 2\,{x}^{2}nb+2\,{x}^{n}an+2\,xny-3\,b{x}^{2}-3\,{x}^{n}a-3\,yx \right ) a}} \right ) +3\,\ln \left ( 9\,{\frac {ab{x}^{2}{n}^{2}+{x}^{n}{a}^{2}{n}^{2}-3\,nab{x}^{2}+c{x}^{2}{n}^{2}-3\,{a}^{2}{x}^{n}n+3\,ab{x}^{2}-3\,cn{x}^{2}+3\,{a}^{2}{x}^{n}+3\,c{x}^{2}}{ \left ( {x}^{2}nb+{x}^{n}an+xny-3\,b{x}^{2}-3\,{x}^{n}a-3\,yx \right ) a}} \right ) -6\,\ln \left ( -9\,{\frac { \left ( a{n}^{2}y-c{n}^{2}x-3\,any+3\,cnx+3\,ya-3\,cx \right ) x}{ \left ( {x}^{n}a+b{x}^{2}+yx \right ) an}} \right ) \right ) }-3\,\ln \left ( ab{x}^{2}+{a}^{2}{x}^{n}+c{x}^{2} \right ) n+6\,n\ln \left ( x \right ) -2\,{\frac {1}{ \left ( n-2 \right ) \left ( n-1 \right ) } \left ( -{n}^{3}\ln \left ( 9\,{\frac {ab{x}^{2}{n}^{2}+{x}^{n}{a}^{2}{n}^{2}-3\,nab{x}^{2}+c{x}^{2}{n}^{2}-3\,{a}^{2}{x}^{n}n+3\,ab{x}^{2}-3\,cn{x}^{2}+3\,{a}^{2}{x}^{n}+3\,c{x}^{2}}{ \left ( {x}^{2}nb+{x}^{n}an+xny-3\,b{x}^{2}-3\,{x}^{n}a-3\,yx \right ) a}} \right ) +\ln \left ( -9\,{\frac { \left ( a{n}^{2}y-c{n}^{2}x-3\,any+3\,cnx+3\,ya-3\,cx \right ) x}{ \left ( {x}^{n}a+b{x}^{2}+yx \right ) an}} \right ) {n}^{3}+{n}^{2}\ln \left ( 9\,{\frac {{x}^{2}{n}^{3}ba+{x}^{n}{n}^{3}{a}^{2}+x{n}^{3}ya-4\,ab{x}^{2}{n}^{2}-4\,{x}^{n}{a}^{2}{n}^{2}-5\,a{n}^{2}xy+6\,nab{x}^{2}+c{x}^{2}{n}^{2}+6\,{a}^{2}{x}^{n}n+9\,axny-3\,ab{x}^{2}-3\,cn{x}^{2}-3\,{a}^{2}{x}^{n}-6\,axy+3\,c{x}^{2}}{ \left ( 2\,{x}^{2}nb+2\,{x}^{n}an+2\,xny-3\,b{x}^{2}-3\,{x}^{n}a-3\,yx \right ) a}} \right ) +4\,{n}^{2}\ln \left ( 9\,{\frac {ab{x}^{2}{n}^{2}+{x}^{n}{a}^{2}{n}^{2}-3\,nab{x}^{2}+c{x}^{2}{n}^{2}-3\,{a}^{2}{x}^{n}n+3\,ab{x}^{2}-3\,cn{x}^{2}+3\,{a}^{2}{x}^{n}+3\,c{x}^{2}}{ \left ( {x}^{2}nb+{x}^{n}an+xny-3\,b{x}^{2}-3\,{x}^{n}a-3\,yx \right ) a}} \right ) -5\,\ln \left ( -9\,{\frac { \left ( a{n}^{2}y-c{n}^{2}x-3\,any+3\,cnx+3\,ya-3\,cx \right ) x}{ \left ( {x}^{n}a+b{x}^{2}+yx \right ) an}} \right ) {n}^{2}-3\,n\ln \left ( 9\,{\frac {{x}^{2}{n}^{3}ba+{x}^{n}{n}^{3}{a}^{2}+x{n}^{3}ya-4\,ab{x}^{2}{n}^{2}-4\,{x}^{n}{a}^{2}{n}^{2}-5\,a{n}^{2}xy+6\,nab{x}^{2}+c{x}^{2}{n}^{2}+6\,{a}^{2}{x}^{n}n+9\,axny-3\,ab{x}^{2}-3\,cn{x}^{2}-3\,{a}^{2}{x}^{n}-6\,axy+3\,c{x}^{2}}{ \left ( 2\,{x}^{2}nb+2\,{x}^{n}an+2\,xny-3\,b{x}^{2}-3\,{x}^{n}a-3\,yx \right ) a}} \right ) -6\,n\ln \left ( 9\,{\frac {ab{x}^{2}{n}^{2}+{x}^{n}{a}^{2}{n}^{2}-3\,nab{x}^{2}+c{x}^{2}{n}^{2}-3\,{a}^{2}{x}^{n}n+3\,ab{x}^{2}-3\,cn{x}^{2}+3\,{a}^{2}{x}^{n}+3\,c{x}^{2}}{ \left ( {x}^{2}nb+{x}^{n}an+xny-3\,b{x}^{2}-3\,{x}^{n}a-3\,yx \right ) a}} \right ) +9\,\ln \left ( -9\,{\frac { \left ( a{n}^{2}y-c{n}^{2}x-3\,any+3\,cnx+3\,ya-3\,cx \right ) x}{ \left ( {x}^{n}a+b{x}^{2}+yx \right ) an}} \right ) n+3\,\ln \left ( 9\,{\frac {{x}^{2}{n}^{3}ba+{x}^{n}{n}^{3}{a}^{2}+x{n}^{3}ya-4\,ab{x}^{2}{n}^{2}-4\,{x}^{n}{a}^{2}{n}^{2}-5\,a{n}^{2}xy+6\,nab{x}^{2}+c{x}^{2}{n}^{2}+6\,{a}^{2}{x}^{n}n+9\,axny-3\,ab{x}^{2}-3\,cn{x}^{2}-3\,{a}^{2}{x}^{n}-6\,axy+3\,c{x}^{2}}{ \left ( 2\,{x}^{2}nb+2\,{x}^{n}an+2\,xny-3\,b{x}^{2}-3\,{x}^{n}a-3\,yx \right ) a}} \right ) +3\,\ln \left ( 9\,{\frac {ab{x}^{2}{n}^{2}+{x}^{n}{a}^{2}{n}^{2}-3\,nab{x}^{2}+c{x}^{2}{n}^{2}-3\,{a}^{2}{x}^{n}n+3\,ab{x}^{2}-3\,cn{x}^{2}+3\,{a}^{2}{x}^{n}+3\,c{x}^{2}}{ \left ( {x}^{2}nb+{x}^{n}an+xny-3\,b{x}^{2}-3\,{x}^{n}a-3\,yx \right ) a}} \right ) -6\,\ln \left ( -9\,{\frac { \left ( a{n}^{2}y-c{n}^{2}x-3\,any+3\,cnx+3\,ya-3\,cx \right ) x}{ \left ( {x}^{n}a+b{x}^{2}+yx \right ) an}} \right ) \right ) }+3\,\ln \left ( ab{x}^{2}+{a}^{2}{x}^{n}+c{x}^{2} \right ) -6\,\ln \left ( x \right ) \right ) } \right ) \]
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Added January 2, 2019.
Problem 2.2.5.43 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ (a y^n+b x^2+c x y) w_x + \left (k y^n+ b x y+c y^2\right ) w_y = 0 \]
Mathematica ✗
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0]; pde = (a*y^n + b*x^2 + c*x*y)*D[w[x, y], x] + (k*y^n + b*x*y + c*y^2)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \text {Failed} \]
Maple ✓
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A'; C:='C'; pde :=(a*y^n+b*x^2+c*x*y)*diff(w(x,y),x)+ ( k*y^n+ b*x*y+c*y^2)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( -1/3\,{\frac {1}{n-2} \left ( 2\,\ln \left ( y \right ) {n}^{2}-\ln \left ( ab{y}^{2}+ck{y}^{2}+{y}^{n}{k}^{2} \right ) {n}^{2}-6\,n\ln \left ( y \right ) -{\frac {n}{ \left ( n-2 \right ) \left ( n-1 \right ) } \left ( -{n}^{3}\ln \left ( 9\,{\frac {ab{y}^{2}{n}^{2}+ck{y}^{2}{n}^{2}+{y}^{n}{k}^{2}{n}^{2}-3\,abn{y}^{2}-3\,ckn{y}^{2}-3\,{y}^{n}{k}^{2}n+3\,ab{y}^{2}+3\,ck{y}^{2}+3\,{y}^{n}{k}^{2}}{ \left ( bnxy+cn{y}^{2}+k{y}^{n}n-3\,yxb-3\,c{y}^{2}-3\,k{y}^{n} \right ) k}} \right ) +\ln \left ( -9\,{\frac { \left ( -a{n}^{2}y+xk{n}^{2}+3\,any-3\,xkn-3\,ya+3\,kx \right ) by}{ \left ( k{y}^{n}+yxb+c{y}^{2} \right ) kn}} \right ) {n}^{3}+{n}^{2}\ln \left ( 9\,{\frac {{n}^{3}ybxk+{n}^{3}{y}^{2}ck+{k}^{2}{y}^{n}{n}^{3}-5\,b{n}^{2}xyk+ab{y}^{2}{n}^{2}-4\,ck{y}^{2}{n}^{2}-4\,{y}^{n}{k}^{2}{n}^{2}+9\,yxbkn-3\,abn{y}^{2}+6\,ckn{y}^{2}+6\,{y}^{n}{k}^{2}n-6\,yxbk+3\,ab{y}^{2}-3\,ck{y}^{2}-3\,{y}^{n}{k}^{2}}{ \left ( 2\,bnxy+2\,cn{y}^{2}+2\,k{y}^{n}n-3\,yxb-3\,c{y}^{2}-3\,k{y}^{n} \right ) k}} \right ) +4\,{n}^{2}\ln \left ( 9\,{\frac {ab{y}^{2}{n}^{2}+ck{y}^{2}{n}^{2}+{y}^{n}{k}^{2}{n}^{2}-3\,abn{y}^{2}-3\,ckn{y}^{2}-3\,{y}^{n}{k}^{2}n+3\,ab{y}^{2}+3\,ck{y}^{2}+3\,{y}^{n}{k}^{2}}{ \left ( bnxy+cn{y}^{2}+k{y}^{n}n-3\,yxb-3\,c{y}^{2}-3\,k{y}^{n} \right ) k}} \right ) -5\,\ln \left ( -9\,{\frac { \left ( -a{n}^{2}y+xk{n}^{2}+3\,any-3\,xkn-3\,ya+3\,kx \right ) by}{ \left ( k{y}^{n}+yxb+c{y}^{2} \right ) kn}} \right ) {n}^{2}-3\,n\ln \left ( 9\,{\frac {{n}^{3}ybxk+{n}^{3}{y}^{2}ck+{k}^{2}{y}^{n}{n}^{3}-5\,b{n}^{2}xyk+ab{y}^{2}{n}^{2}-4\,ck{y}^{2}{n}^{2}-4\,{y}^{n}{k}^{2}{n}^{2}+9\,yxbkn-3\,abn{y}^{2}+6\,ckn{y}^{2}+6\,{y}^{n}{k}^{2}n-6\,yxbk+3\,ab{y}^{2}-3\,ck{y}^{2}-3\,{y}^{n}{k}^{2}}{ \left ( 2\,bnxy+2\,cn{y}^{2}+2\,k{y}^{n}n-3\,yxb-3\,c{y}^{2}-3\,k{y}^{n} \right ) k}} \right ) -6\,n\ln \left ( 9\,{\frac {ab{y}^{2}{n}^{2}+ck{y}^{2}{n}^{2}+{y}^{n}{k}^{2}{n}^{2}-3\,abn{y}^{2}-3\,ckn{y}^{2}-3\,{y}^{n}{k}^{2}n+3\,ab{y}^{2}+3\,ck{y}^{2}+3\,{y}^{n}{k}^{2}}{ \left ( bnxy+cn{y}^{2}+k{y}^{n}n-3\,yxb-3\,c{y}^{2}-3\,k{y}^{n} \right ) k}} \right ) +9\,\ln \left ( -9\,{\frac { \left ( -a{n}^{2}y+xk{n}^{2}+3\,any-3\,xkn-3\,ya+3\,kx \right ) by}{ \left ( k{y}^{n}+yxb+c{y}^{2} \right ) kn}} \right ) n+3\,\ln \left ( 9\,{\frac {{n}^{3}ybxk+{n}^{3}{y}^{2}ck+{k}^{2}{y}^{n}{n}^{3}-5\,b{n}^{2}xyk+ab{y}^{2}{n}^{2}-4\,ck{y}^{2}{n}^{2}-4\,{y}^{n}{k}^{2}{n}^{2}+9\,yxbkn-3\,abn{y}^{2}+6\,ckn{y}^{2}+6\,{y}^{n}{k}^{2}n-6\,yxbk+3\,ab{y}^{2}-3\,ck{y}^{2}-3\,{y}^{n}{k}^{2}}{ \left ( 2\,bnxy+2\,cn{y}^{2}+2\,k{y}^{n}n-3\,yxb-3\,c{y}^{2}-3\,k{y}^{n} \right ) k}} \right ) +3\,\ln \left ( 9\,{\frac {ab{y}^{2}{n}^{2}+ck{y}^{2}{n}^{2}+{y}^{n}{k}^{2}{n}^{2}-3\,abn{y}^{2}-3\,ckn{y}^{2}-3\,{y}^{n}{k}^{2}n+3\,ab{y}^{2}+3\,ck{y}^{2}+3\,{y}^{n}{k}^{2}}{ \left ( bnxy+cn{y}^{2}+k{y}^{n}n-3\,yxb-3\,c{y}^{2}-3\,k{y}^{n} \right ) k}} \right ) -6\,\ln \left ( -9\,{\frac { \left ( -a{n}^{2}y+xk{n}^{2}+3\,any-3\,xkn-3\,ya+3\,kx \right ) by}{ \left ( k{y}^{n}+yxb+c{y}^{2} \right ) kn}} \right ) \right ) }+3\,\ln \left ( ab{y}^{2}+ck{y}^{2}+{y}^{n}{k}^{2} \right ) n+6\,\ln \left ( y \right ) +2\,{\frac {1}{ \left ( n-2 \right ) \left ( n-1 \right ) } \left ( -{n}^{3}\ln \left ( 9\,{\frac {ab{y}^{2}{n}^{2}+ck{y}^{2}{n}^{2}+{y}^{n}{k}^{2}{n}^{2}-3\,abn{y}^{2}-3\,ckn{y}^{2}-3\,{y}^{n}{k}^{2}n+3\,ab{y}^{2}+3\,ck{y}^{2}+3\,{y}^{n}{k}^{2}}{ \left ( bnxy+cn{y}^{2}+k{y}^{n}n-3\,yxb-3\,c{y}^{2}-3\,k{y}^{n} \right ) k}} \right ) +\ln \left ( -9\,{\frac { \left ( -a{n}^{2}y+xk{n}^{2}+3\,any-3\,xkn-3\,ya+3\,kx \right ) by}{ \left ( k{y}^{n}+yxb+c{y}^{2} \right ) kn}} \right ) {n}^{3}+{n}^{2}\ln \left ( 9\,{\frac {{n}^{3}ybxk+{n}^{3}{y}^{2}ck+{k}^{2}{y}^{n}{n}^{3}-5\,b{n}^{2}xyk+ab{y}^{2}{n}^{2}-4\,ck{y}^{2}{n}^{2}-4\,{y}^{n}{k}^{2}{n}^{2}+9\,yxbkn-3\,abn{y}^{2}+6\,ckn{y}^{2}+6\,{y}^{n}{k}^{2}n-6\,yxbk+3\,ab{y}^{2}-3\,ck{y}^{2}-3\,{y}^{n}{k}^{2}}{ \left ( 2\,bnxy+2\,cn{y}^{2}+2\,k{y}^{n}n-3\,yxb-3\,c{y}^{2}-3\,k{y}^{n} \right ) k}} \right ) +4\,{n}^{2}\ln \left ( 9\,{\frac {ab{y}^{2}{n}^{2}+ck{y}^{2}{n}^{2}+{y}^{n}{k}^{2}{n}^{2}-3\,abn{y}^{2}-3\,ckn{y}^{2}-3\,{y}^{n}{k}^{2}n+3\,ab{y}^{2}+3\,ck{y}^{2}+3\,{y}^{n}{k}^{2}}{ \left ( bnxy+cn{y}^{2}+k{y}^{n}n-3\,yxb-3\,c{y}^{2}-3\,k{y}^{n} \right ) k}} \right ) -5\,\ln \left ( -9\,{\frac { \left ( -a{n}^{2}y+xk{n}^{2}+3\,any-3\,xkn-3\,ya+3\,kx \right ) by}{ \left ( k{y}^{n}+yxb+c{y}^{2} \right ) kn}} \right ) {n}^{2}-3\,n\ln \left ( 9\,{\frac {{n}^{3}ybxk+{n}^{3}{y}^{2}ck+{k}^{2}{y}^{n}{n}^{3}-5\,b{n}^{2}xyk+ab{y}^{2}{n}^{2}-4\,ck{y}^{2}{n}^{2}-4\,{y}^{n}{k}^{2}{n}^{2}+9\,yxbkn-3\,abn{y}^{2}+6\,ckn{y}^{2}+6\,{y}^{n}{k}^{2}n-6\,yxbk+3\,ab{y}^{2}-3\,ck{y}^{2}-3\,{y}^{n}{k}^{2}}{ \left ( 2\,bnxy+2\,cn{y}^{2}+2\,k{y}^{n}n-3\,yxb-3\,c{y}^{2}-3\,k{y}^{n} \right ) k}} \right ) -6\,n\ln \left ( 9\,{\frac {ab{y}^{2}{n}^{2}+ck{y}^{2}{n}^{2}+{y}^{n}{k}^{2}{n}^{2}-3\,abn{y}^{2}-3\,ckn{y}^{2}-3\,{y}^{n}{k}^{2}n+3\,ab{y}^{2}+3\,ck{y}^{2}+3\,{y}^{n}{k}^{2}}{ \left ( bnxy+cn{y}^{2}+k{y}^{n}n-3\,yxb-3\,c{y}^{2}-3\,k{y}^{n} \right ) k}} \right ) +9\,\ln \left ( -9\,{\frac { \left ( -a{n}^{2}y+xk{n}^{2}+3\,any-3\,xkn-3\,ya+3\,kx \right ) by}{ \left ( k{y}^{n}+yxb+c{y}^{2} \right ) kn}} \right ) n+3\,\ln \left ( 9\,{\frac {{n}^{3}ybxk+{n}^{3}{y}^{2}ck+{k}^{2}{y}^{n}{n}^{3}-5\,b{n}^{2}xyk+ab{y}^{2}{n}^{2}-4\,ck{y}^{2}{n}^{2}-4\,{y}^{n}{k}^{2}{n}^{2}+9\,yxbkn-3\,abn{y}^{2}+6\,ckn{y}^{2}+6\,{y}^{n}{k}^{2}n-6\,yxbk+3\,ab{y}^{2}-3\,ck{y}^{2}-3\,{y}^{n}{k}^{2}}{ \left ( 2\,bnxy+2\,cn{y}^{2}+2\,k{y}^{n}n-3\,yxb-3\,c{y}^{2}-3\,k{y}^{n} \right ) k}} \right ) +3\,\ln \left ( 9\,{\frac {ab{y}^{2}{n}^{2}+ck{y}^{2}{n}^{2}+{y}^{n}{k}^{2}{n}^{2}-3\,abn{y}^{2}-3\,ckn{y}^{2}-3\,{y}^{n}{k}^{2}n+3\,ab{y}^{2}+3\,ck{y}^{2}+3\,{y}^{n}{k}^{2}}{ \left ( bnxy+cn{y}^{2}+k{y}^{n}n-3\,yxb-3\,c{y}^{2}-3\,k{y}^{n} \right ) k}} \right ) -6\,\ln \left ( -9\,{\frac { \left ( -a{n}^{2}y+xk{n}^{2}+3\,any-3\,xkn-3\,ya+3\,kx \right ) by}{ \left ( k{y}^{n}+yxb+c{y}^{2} \right ) kn}} \right ) \right ) }-3\,\ln \left ( ab{y}^{2}+ck{y}^{2}+{y}^{n}{k}^{2} \right ) \right ) } \right ) \]
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Added January 2, 2019.
Problem 2.2.5.44 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ (a x^n + b x^m + c) w_x + \left (c y^2-b x^{m-1} y+ a x^{n-2}\right ) w_y = 0 \]
Mathematica ✗
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0]; pde = (a*x^n + b*x^m + c)*D[w[x, y], x] + (c*y^2 - b*x^(m - 1)*y + a*x^(n - 2))*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \text {Failed} \]
Maple ✗
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A'; C:='C'; pde :=(a*x^n + b*x^m + c)*diff(w(x,y),x)+ ( c*y^2-b*x^(m-1)*y+ a*x^(n-2))*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ \text { sol=() } \]
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Added January 2, 2019.
Problem 2.2.5.45 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ (a x^n + b x^m + c) w_x + \left (a x^{n-2} y^2 + b x^{m-1} y + c \right ) w_y = 0 \]
Mathematica ✗
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0]; pde = (a*x^n + b*x^m + c)*D[w[x, y], x] + (a*x^(n - 2)*y^2 + b*x^(m - 1)*y + c)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \text {Failed} \]
Maple ✗
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A'; C:='C'; pde :=(a*x^n + b*x^m + c)*diff(w(x,y),x)+ ( a*x^(n-2)*y^2 + b*x^(m-1)*y + c)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ \text { sol=() } \]
____________________________________________________________________________________
Added January 2, 2019.
Problem 2.2.5.46 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ (a x^n + b x^m + c) w_x + \left ( \alpha x^k y^2 + \beta x^s y - \alpha \lambda ^2 x^k + \beta \lambda x^s \right ) w_y = 0 \]
Mathematica ✗
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda]; pde = (a*x^n + b*x^m + c)*D[w[x, y], x] + (alpha*x^k*y^2 + beta*x^s*y - alpha*lambda^2*x^k + beta*lambda*x^s)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \text {\$Aborted} \] Timed out
Maple ✓
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A'; C:='C';lambda:='lambda';s:='s'; pde :=(a*x^n + b*x^m + c)*diff(w(x,y),x)+ (alpha*x^k*y^2 + beta*x^s*y - alpha*lambda^2*x^k + beta*lambda*x^s)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{1 \left ( \int \!{\frac {{x}^{k}\alpha }{{x}^{n}a+{x}^{m}b+c}{{\rm e}^{\int \!{\frac {-2\,{x}^{k}\alpha \,\lambda +{x}^{s}\beta }{{x}^{n}a+{x}^{m}b+c}}\,{\rm d}x}}}\,{\rm d}x\lambda \,{{\rm e}^{\int \!{\frac {-2\,{x}^{k}\alpha \,\lambda +{x}^{s}\beta }{{x}^{n}a+{x}^{m}b+c}}\,{\rm d}x+\int \!-{\frac {-2\,{x}^{k}\alpha \,\lambda +{x}^{s}\beta }{{x}^{n}a+{x}^{m}b+c}}\,{\rm d}x}}+y\int \!{\frac {{x}^{k}\alpha }{{x}^{n}a+{x}^{m}b+c}{{\rm e}^{\int \!{\frac {-2\,{x}^{k}\alpha \,\lambda +{x}^{s}\beta }{{x}^{n}a+{x}^{m}b+c}}\,{\rm d}x}}}\,{\rm d}x+{{\rm e}^{\int \!{\frac {-2\,{x}^{k}\alpha \,\lambda +{x}^{s}\beta }{{x}^{n}a+{x}^{m}b+c}}\,{\rm d}x}} \right ) \left ( {{\rm e}^{\int \!{\frac {-2\,{x}^{k}\alpha \,\lambda +{x}^{s}\beta }{{x}^{n}a+{x}^{m}b+c}}\,{\rm d}x+\int \!-{\frac {-2\,{x}^{k}\alpha \,\lambda +{x}^{s}\beta }{{x}^{n}a+{x}^{m}b+c}}\,{\rm d}x}}\lambda +y \right ) ^{-1}} \right ) \]
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Added January 2, 2019.
Problem 2.2.5.47 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ x(a x^n + b x^m + c) w_x - \left ( s x^k y^2 -(a x^n + b x^m+c) y - s \lambda x^{k+2} \right ) w_y = 0 \]
Mathematica ✓
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda]; pde = x*(a*x^n + b*x^m + c)*D[w[x, y], x] - (s*x^k*y^2 - (a*x^n + b*x^m + c)*y - s*lambda*x^(k + 2))*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to c_1\left (\frac {\tanh ^{-1}\left (\frac {y}{\sqrt {\lambda } x}\right )-\sqrt {\lambda } \int _1^x \frac {s K[1]^k}{a K[1]^n+b K[1]^m+c} \, dK[1]}{\sqrt {\lambda }}\right )\right \}\right \} \]
Maple ✓
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A'; C:='C';lambda:='lambda';s:='s'; pde :=x*(a*x^n + b*x^m + c)*diff(w(x,y),x)- (s*x^k*y^2 -(a*x^n + b*x^m+c)*y - s*lambda*x^(k+2))*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1}{s\sqrt {\lambda }} \left ( -\int \!{\frac {{x}^{k}}{{x}^{n}a+{x}^{m}b+c}}\,{\rm d}xs\sqrt {\lambda }+\arctanh \left ( {\frac {y}{x\sqrt {\lambda }}} \right ) \right ) } \right ) \]
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Added January 2, 2019.
Problem 2.2.5.48 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ (a x^n + b x^m + c) w_x + \left ( (a x^n+b x^m + c)y^2-a n(n-1)x^{n-2}-b m(m-1) x^{m-2}\right ) w_y = 0 \]
Mathematica ✗
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda]; pde = (a*x^n + b*x^m + c)*D[w[x, y], x] + ((a*x^n + b*x^m + c)*y^2 - a*n*(n - 1)*x^(n - 2) - b*m*(m - 1)*x^(m - 2))*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \text {Failed} \]
Maple ✓
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A'; C:='C';lambda:='lambda';s:='s'; pde :=(a*x^n + b*x^m + c)*diff(w(x,y),x)+ ((a*x^n+b*x^m + c)*y^2-a*n*(n-1)*x^(n-2)-b*m*(m-1)*x^(m-2))*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac {1}{{x}^{n}acn+{x}^{m}bcm+{x}^{1+2\,n}y{a}^{2}+y{x}^{2\,m+1}{b}^{2}+{a}^{2}{x}^{2\,n}n+{x}^{2\,m}{b}^{2}m+{c}^{2}xy+2\,a{x}^{m+n+1}yb+2\,cya{x}^{n+1}+2\,y{x}^{m+1}bc+a{x}^{n+m}bm+a{x}^{n+m}bn} \left ( x+ \left ( -{\frac {x}{ \left ( bm{x}^{m}-{x}^{m}bn-cn \right ) \left ( {x}^{n}a+{x}^{m}b+c \right ) }}+\int \!-{\frac {b{m}^{2}{x}^{m}-{x}^{m}b{n}^{2}-bm{x}^{m}+{x}^{m}bn-c{n}^{2}+cn}{ \left ( bm{x}^{m}-{x}^{m}bn-cn \right ) ^{2} \left ( {x}^{n}a+{x}^{m}b+c \right ) }}\,{\rm d}x \right ) {x}^{1+2\,n}y{a}^{2}+ \left ( -{\frac {x}{ \left ( bm{x}^{m}-{x}^{m}bn-cn \right ) \left ( {x}^{n}a+{x}^{m}b+c \right ) }}+\int \!-{\frac {b{m}^{2}{x}^{m}-{x}^{m}b{n}^{2}-bm{x}^{m}+{x}^{m}bn-c{n}^{2}+cn}{ \left ( bm{x}^{m}-{x}^{m}bn-cn \right ) ^{2} \left ( {x}^{n}a+{x}^{m}b+c \right ) }}\,{\rm d}x \right ) y{x}^{2\,m+1}{b}^{2}+ \left ( -{\frac {x}{ \left ( bm{x}^{m}-{x}^{m}bn-cn \right ) \left ( {x}^{n}a+{x}^{m}b+c \right ) }}+\int \!-{\frac {b{m}^{2}{x}^{m}-{x}^{m}b{n}^{2}-bm{x}^{m}+{x}^{m}bn-c{n}^{2}+cn}{ \left ( bm{x}^{m}-{x}^{m}bn-cn \right ) ^{2} \left ( {x}^{n}a+{x}^{m}b+c \right ) }}\,{\rm d}x \right ) y{c}^{2}x+ \left ( -{\frac {x}{ \left ( bm{x}^{m}-{x}^{m}bn-cn \right ) \left ( {x}^{n}a+{x}^{m}b+c \right ) }}+\int \!-{\frac {b{m}^{2}{x}^{m}-{x}^{m}b{n}^{2}-bm{x}^{m}+{x}^{m}bn-c{n}^{2}+cn}{ \left ( bm{x}^{m}-{x}^{m}bn-cn \right ) ^{2} \left ( {x}^{n}a+{x}^{m}b+c \right ) }}\,{\rm d}x \right ) {x}^{2\,n}{a}^{2}n+ \left ( -{\frac {x}{ \left ( bm{x}^{m}-{x}^{m}bn-cn \right ) \left ( {x}^{n}a+{x}^{m}b+c \right ) }}+\int \!-{\frac {b{m}^{2}{x}^{m}-{x}^{m}b{n}^{2}-bm{x}^{m}+{x}^{m}bn-c{n}^{2}+cn}{ \left ( bm{x}^{m}-{x}^{m}bn-cn \right ) ^{2} \left ( {x}^{n}a+{x}^{m}b+c \right ) }}\,{\rm d}x \right ) {x}^{2\,m}{b}^{2}m+ \left ( -{\frac {x}{ \left ( bm{x}^{m}-{x}^{m}bn-cn \right ) \left ( {x}^{n}a+{x}^{m}b+c \right ) }}+\int \!-{\frac {b{m}^{2}{x}^{m}-{x}^{m}b{n}^{2}-bm{x}^{m}+{x}^{m}bn-c{n}^{2}+cn}{ \left ( bm{x}^{m}-{x}^{m}bn-cn \right ) ^{2} \left ( {x}^{n}a+{x}^{m}b+c \right ) }}\,{\rm d}x \right ) {x}^{n}acn+ \left ( -{\frac {x}{ \left ( bm{x}^{m}-{x}^{m}bn-cn \right ) \left ( {x}^{n}a+{x}^{m}b+c \right ) }}+\int \!-{\frac {b{m}^{2}{x}^{m}-{x}^{m}b{n}^{2}-bm{x}^{m}+{x}^{m}bn-c{n}^{2}+cn}{ \left ( bm{x}^{m}-{x}^{m}bn-cn \right ) ^{2} \left ( {x}^{n}a+{x}^{m}b+c \right ) }}\,{\rm d}x \right ) {x}^{m}bcm+2\, \left ( -{\frac {x}{ \left ( bm{x}^{m}-{x}^{m}bn-cn \right ) \left ( {x}^{n}a+{x}^{m}b+c \right ) }}+\int \!-{\frac {b{m}^{2}{x}^{m}-{x}^{m}b{n}^{2}-bm{x}^{m}+{x}^{m}bn-c{n}^{2}+cn}{ \left ( bm{x}^{m}-{x}^{m}bn-cn \right ) ^{2} \left ( {x}^{n}a+{x}^{m}b+c \right ) }}\,{\rm d}x \right ) {x}^{m+n+1}yab+2\, \left ( -{\frac {x}{ \left ( bm{x}^{m}-{x}^{m}bn-cn \right ) \left ( {x}^{n}a+{x}^{m}b+c \right ) }}+\int \!-{\frac {b{m}^{2}{x}^{m}-{x}^{m}b{n}^{2}-bm{x}^{m}+{x}^{m}bn-c{n}^{2}+cn}{ \left ( bm{x}^{m}-{x}^{m}bn-cn \right ) ^{2} \left ( {x}^{n}a+{x}^{m}b+c \right ) }}\,{\rm d}x \right ) {x}^{n+1}yac+2\, \left ( -{\frac {x}{ \left ( bm{x}^{m}-{x}^{m}bn-cn \right ) \left ( {x}^{n}a+{x}^{m}b+c \right ) }}+\int \!-{\frac {b{m}^{2}{x}^{m}-{x}^{m}b{n}^{2}-bm{x}^{m}+{x}^{m}bn-c{n}^{2}+cn}{ \left ( bm{x}^{m}-{x}^{m}bn-cn \right ) ^{2} \left ( {x}^{n}a+{x}^{m}b+c \right ) }}\,{\rm d}x \right ) y{x}^{m+1}bc+ \left ( -{\frac {x}{ \left ( bm{x}^{m}-{x}^{m}bn-cn \right ) \left ( {x}^{n}a+{x}^{m}b+c \right ) }}+\int \!-{\frac {b{m}^{2}{x}^{m}-{x}^{m}b{n}^{2}-bm{x}^{m}+{x}^{m}bn-c{n}^{2}+cn}{ \left ( bm{x}^{m}-{x}^{m}bn-cn \right ) ^{2} \left ( {x}^{n}a+{x}^{m}b+c \right ) }}\,{\rm d}x \right ) {x}^{n+m}abm+ \left ( -{\frac {x}{ \left ( bm{x}^{m}-{x}^{m}bn-cn \right ) \left ( {x}^{n}a+{x}^{m}b+c \right ) }}+\int \!-{\frac {b{m}^{2}{x}^{m}-{x}^{m}b{n}^{2}-bm{x}^{m}+{x}^{m}bn-c{n}^{2}+cn}{ \left ( bm{x}^{m}-{x}^{m}bn-cn \right ) ^{2} \left ( {x}^{n}a+{x}^{m}b+c \right ) }}\,{\rm d}x \right ) {x}^{n+m}abn \right ) } \right ) \]
____________________________________________________________________________________
Added January 2, 2019.
Problem 2.2.5.49 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ (a x^n + b y^n + x) w_x + \left ( \alpha x^k y^{n-k} + \beta x^m y^{n-m} + y \right ) w_y = 0 \]
Mathematica ✗
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda]; pde = (a*x^n + b*y^n + x)*D[w[x, y], x] + (alpha*x^k*y^(n - k) + beta*x^m*y^(n - m) + y)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \text {Failed} \]
Maple ✗
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A'; C:='C';lambda:='lambda';s:='s'; pde := (a*x^n + b*y^n + x)*diff(w(x,y),x)+ (alpha*x^k*y^(n-k) +beta*x^m*y^(n-m) + y )*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ \text { sol=() } \]
____________________________________________________________________________________
Added January 2, 2019.
Problem 2.2.5.50 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ (a x^n + b y^n + A x^2 + B x y) w_x + \left ( \alpha x^k y^{n-k} + \beta x^m y^{n-m} + A x y + B y^2\right ) w_y = 0 \]
Mathematica ✗
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B]; pde = (a*x^n + b*y^n + A*x^2 + B*x*y)*D[w[x, y], x] + (alpha*x^k*y^(n - k) + beta*x^m*y^(n - m) + A*x*y + B*y^2)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \text {Failed} \]
Maple ✗
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A'; C:='C';lambda:='lambda';s:='s';B:='B'; pde := (a*x^n + b*y^n + A*x^2 + B*x*y)*diff(w(x,y),x)+ (alpha*x^k*y^(n-k)+beta*x^m*y^(n-m) + A*x*y + B*y^2 )*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ \text { sol=() } \]
____________________________________________________________________________________
Added January 2, 2019.
Problem 2.2.5.51 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ (a y^m + b x^n + s) w_x - \left ( \alpha x^k + b n x^{n-1} y + \beta \right ) w_y = 0 \]
Mathematica ✗
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s]; pde = (a*y^m + b*x^n + s)*D[w[x, y], x] - (alpha*x^k + b*n*x^(n - 1)*y + beta)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \text {Failed} \]
Maple ✓
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A'; C:='C';lambda:='lambda';s:='s';B:='B'; pde := (a*y^m + b*x^n + s)*diff(w(x,y),x)- (alpha*x^k + b*n*x^(n-1)* y + beta)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac {{x}^{n}bkmy+{x}^{k}x\alpha \,m+{x}^{n}bky+bym{x}^{n}+\beta \,kmx+kmsy+a{y}^{m}yk+{x}^{k}x\alpha +{x}^{n}by+\beta \,kx+\beta \,mx+ksy+msy+a{y}^{m}y+\beta \,x+sy}{km+k+m+1}} \right ) \]
____________________________________________________________________________________
Added January 2, 2019.
Problem 2.2.5.52 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ (a x^n y^m +x) w_x + \left ( b x^k y^{n+m-k} + y \right ) w_y = 0 \]
Mathematica ✗
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s]; pde = (a*x^n*y^m + x)*D[w[x, y], x] + (b*x^k*y^(n + m - k) + y)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \text {Failed} \]
Maple ✗
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A'; C:='C';lambda:='lambda';s:='s';B:='B'; pde := (a*x^n*y^m +x)*diff(w(x,y),x)+ (b*x^k*y^(n+m-k) + y)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ \text { sol=() } \]
____________________________________________________________________________________
Added January 2, 2019.
Problem 2.2.5.53 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ x(a x^n y^m +\alpha ) w_x - y \left ( b x^n y^m + \beta \right ) w_y = 0 \]
Mathematica ✗
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s]; pde = x*(a*x^n*y^m + alpha)*D[w[x, y], x] - y*(b*x^n*y^m + beta)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \text {Failed} \]
Maple ✓
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A'; C:='C';lambda:='lambda';s:='s';B:='B'; pde := x*(a*x^n*y^m +alpha)*diff(w(x,y),x)- y*( b*x^n*y^m + beta )*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( \left ( {y}^{m} \right ) ^{\alpha \, \left ( an-bm \right ) }{x}^{\beta \,m \left ( an-bm \right ) } \left ( {x}^{n}{y}^{m}an-b{x}^{n}{y}^{m}m+\alpha \,n-\beta \,m \right ) ^{-m \left ( a\beta -\alpha \,b \right ) } \right ) \]
____________________________________________________________________________________
Added January 2, 2019.
Problem 2.2.5.54 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ x(a n x^k y^{n+k} + s) w_x - y \left ( b m x^{m+k} y^k + s \right ) w_y = 0 \]
Mathematica ✗
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s]; pde = x*(a*n*x^k*y^(n + k) + s)*D[w[x, y], x] - y*(b*m*x^(m + k)*y^k + s)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \text {Failed} \]
Maple ✗
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A'; C:='C';lambda:='lambda';s:='s';B:='B'; pde := x*(a*n*x^k*y^(n+k) + s)*diff(w(x,y),x)- y*( b*m*x^(m+k)*y^k + s )*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ \text { sol=() } \]
____________________________________________________________________________________
Added January 2, 2019.
Problem 2.2.5.55 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ (a x^n y^m + A x^2 + B x y) w_x + \left ( b x^k y^{n+m-k} + A x y+ B y^2 \right ) w_y = 0 \]
Mathematica ✗
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s]; pde = (a*x^n*y^m + A*x^2 + B*x*y)*D[w[x, y], x] + (b*x^k*y^(n + m - k) + A*x*y + B*y^2)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \text {Failed} \]
Maple ✗
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A'; C:='C';lambda:='lambda';s:='s';B:='B'; pde := (a*x^n*y^m + A*x^2 + B*x*y)*diff(w(x,y),x)+ (b*x^k*y^(n+m-k) + A*x*y+ B*y^2)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ \text { sol=() } \]
____________________________________________________________________________________
Added January 2, 2019.
Problem 2.2.5.56 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ (a x^n y^m + b x y^k) w_x + \left ( \alpha y^s + \beta \right ) w_y = 0 \]
Mathematica ✗
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s]; pde = (a*x^n*y^m + b*x*y^k)*D[w[x, y], x] + (alpha*y^s + beta)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \text {Failed} \] Timed out
Maple ✓
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A'; C:='C';lambda:='lambda';s:='s';B:='B'; pde := (a*x^n*y^m + b*x*y^k)*diff(w(x,y),x)+ (alpha*y^s + beta)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( {x}^{-n+1}{{\rm e}^{b \left ( n-1 \right ) \int \!{\frac {{y}^{k}}{\alpha \,{y}^{s}+\beta }}\,{\rm d}y}}+an\int \!{\frac {{y}^{m}}{\alpha \,{y}^{s}+\beta }{{\rm e}^{b \left ( n-1 \right ) \int \!{\frac {{y}^{k}}{\alpha \,{y}^{s}+\beta }}\,{\rm d}y}}}\,{\rm d}y-a\int \!{\frac {{y}^{m}}{\alpha \,{y}^{s}+\beta }{{\rm e}^{b \left ( n-1 \right ) \int \!{\frac {{y}^{k}}{\alpha \,{y}^{s}+\beta }}\,{\rm d}y}}}\,{\rm d}y \right ) \]