Added Feb. 9, 2019.
Problem Chapter 3.2.4.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x +b w_y = c x^n + d y^m \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*D[w[x, y], y] == c*x^n + d*y^m; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (y-\frac {b x}{a}\right )+\frac {c x^{n+1}}{a n+a}+\frac {d y^{m+1}}{b m+b}\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x) + b*diff(w(x,y),y) =c*x^n+d*y^m; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[w \left ( x,y \right ) ={\frac {1}{b \left ( m+1 \right ) a \left ( n+1 \right ) } \left ( b \left ( m+1 \right ) a \left ( n+1 \right ) {\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) +bc \left ( m+1 \right ) {x}^{n+1}+d{y}^{m+1}a \left ( n+1 \right ) \right ) }\]
____________________________________________________________________________________
Added Feb. 9, 2019.
Problem Chapter 3.2.4.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x +b w_y = c x^n y \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*D[w[x, y], y] == c*x^n*y; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \frac {a^2 \left (n^2+3 n+2\right ) c_1\left (y-\frac {b x}{a}\right )-c x^{n+1} (b x-a (n+2) y)}{a^2 (n+1) (n+2)}\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x) + b*diff(w(x,y),y) =c*x^n*y; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[w \left ( x,y \right ) ={\frac {1}{ \left ( n+2 \right ) \left ( n+1 \right ) {a}^{2}} \left ( \left ( n+2 \right ) \left ( n+1 \right ) {a}^{2}{\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) +{x}^{n+1}c \left ( y \left ( n+2 \right ) a-bx \right ) \right ) }\]
____________________________________________________________________________________
Added Feb. 9, 2019.
Problem Chapter 3.2.4.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ x w_x +y w_y = a(x^2+y^2)^k \]
Mathematica ✓
ClearAll["Global`*"]; pde = x*D[w[x, y], x] + y*D[w[x, y], y] == a*(x^2 + y^2)^k; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (\frac {y}{x}\right )+\frac {a \left (x^2+y^2\right )^k}{2 k}\right \}\right \}\]
Maple ✓
restart; pde :=x*diff(w(x,y),x) + y*diff(w(x,y),y) =a*(x^2+y^2)^k; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[w \left ( x,y \right ) =1/2\,{\frac {1}{k} \left ( a \left ( {x}^{2}+{y}^{2} \right ) ^{k}+2\,{\it \_F1} \left ( {\frac {y}{x}} \right ) k \right ) }\]
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Added Feb. 9, 2019.
Problem Chapter 3.2.4.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a x w_x +b y w_y = c x^n y^m \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*x*D[w[x, y], x] + b*y*D[w[x, y], y] == c*x^n*y^m; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (y x^{-\frac {b}{a}}\right )+\frac {c y^m x^n}{a n+b m}\right \}\right \}\]
Maple ✓
restart; pde :=a*x*diff(w(x,y),x) + b*y*diff(w(x,y),y) =c*x^n*y^m; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[w \left ( x,y \right ) ={\frac {c{x}^{n}{y}^{m}}{an+bm}}+{\it \_F1} \left ( y{x}^{-{\frac {b}{a}}} \right ) \]
____________________________________________________________________________________
Added Feb. 9, 2019.
Problem Chapter 3.2.4.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a x w_x +b y w_y = c x^n + d y^m \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*x*D[w[x, y], x] + b*y*D[w[x, y], y] == c*x^n + d*y^m; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (y x^{-\frac {b}{a}}\right )+\frac {c x^n}{a n}+\frac {d y^m}{b m}\right \}\right \}\]
Maple ✓
restart; pde :=a*x*diff(w(x,y),x) + b*y*diff(w(x,y),y) =c*x^n + d*y^m; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[w \left ( x,y \right ) =\int ^{x}\!{\frac {1}{a{\it \_a}} \left ( c{{\it \_a}}^{n}+d \left ( y{x}^{-{\frac {b}{a}}}{{\it \_a}}^{{\frac {b}{a}}} \right ) ^{m} \right ) }{d{\it \_a}}+{\it \_F1} \left ( y{x}^{-{\frac {b}{a}}} \right ) \] Result has unresolved integral
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Added Feb. 9, 2019.
Problem Chapter 3.2.4.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ m x w_x +n y w_y = (a x^n+ b y^m)^k \]
Mathematica ✓
ClearAll["Global`*"]; pde = m*x*D[w[x, y], x] + n*y*D[w[x, y], y] == (a*x^n + b*y^m)^k; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (y x^{-\frac {n}{m}}\right )+\frac {\left (a x^n+b y^m\right )^k}{k m n}\right \}\right \}\]
Maple ✓
restart; pde :=m*x*diff(w(x,y),x) + n*y*diff(w(x,y),y) =(a*x^n+b*y^m)^k; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[w \left ( x,y \right ) ={\frac {1}{knm} \left ( {\it \_F1} \left ( y{x}^{-{\frac {n}{m}}} \right ) knm+ \left ( {x}^{n}a+{y}^{m}b \right ) ^{k} \right ) }\] Result has unresolved integral
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Added Feb. 9, 2019.
Problem Chapter 3.2.4.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a x^n w_x +b y^m w_y = c x^k+ d y^s \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*x^n*D[w[x, y], x] + b*y^m*D[w[x, y], y] == c*x^k + d*y^s; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (\frac {b x^{1-n}}{a (n-1)}-\frac {y^{1-m}}{m-1}\right )+\frac {c x^{k-n+1}}{a k-a n+a}-\frac {d y^{1-m} \left (\left (y^{m-1}\right )^{\frac {1}{m-1}}\right )^s}{b (m-s-1)}\right \}\right \}\]
Maple ✓
restart; pde :=a*x^n*diff(w(x,y),x) + n*y^m*diff(w(x,y),y) =c*x^k+d*y^s; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[w \left ( x,y \right ) =-{\frac {1}{ \left ( m-s-1 \right ) n \left ( k-n+1 \right ) a} \left ( {a}^{2}{y}^{-m+1}{a}^{{\frac {s}{m-1}}-1} \left ( n-1 \right ) ^{{\frac {s}{m-1}}} \left ( a \left ( n-1 \right ) {y}^{-m+1} \right ) ^{-{\frac {s}{m-1}}}d \left ( k-n+1 \right ) {{\rm e}^{{\frac {-i\pi \,s}{2\,m-2} \left ( {\it csgn} \left ( i{y}^{-m+1}a \right ) \left ( {\it csgn} \left ( i{y}^{-m+1}a \right ) -{\it csgn} \left ( {\frac {i}{n-1}} \right ) \right ) {\it csgn} \left ( ia \left ( n-1 \right ) {y}^{-m+1} \right ) - \left ( {\it csgn} \left ( i{y}^{-m+1}a \right ) \right ) ^{3}+ \left ( {\it csgn} \left ( i{y}^{-m+1}a \right ) \right ) ^{2}{\it csgn} \left ( {\frac {i}{n-1}} \right ) +{\it csgn} \left ( i{y}^{-m+1} \right ) \left ( {\it csgn} \left ( i{y}^{-m+1} \right ) -{\it csgn} \left ( {\frac {i}{a}} \right ) \right ) {\it csgn} \left ( i{y}^{-m+1}a \right ) - \left ( {\it csgn} \left ( i{y}^{-m+1} \right ) \right ) ^{2} \left ( {\it csgn} \left ( i{y}^{-m+1} \right ) -{\it csgn} \left ( {\frac {i}{a}} \right ) \right ) \right ) }}}- \left ( m-s-1 \right ) n \left ( a \left ( k-n+1 \right ) {\it \_F1} \left ( {\frac {-{x}^{-n+1}n \left ( m-1 \right ) +a \left ( n-1 \right ) {y}^{-m+1}}{a \left ( n-1 \right ) }} \right ) +{x}^{k-n+1}c \right ) \right ) }\] Result has unresolved integral
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Added Feb. 9, 2019.
Problem Chapter 3.2.4.8 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a x^n w_x +b x^m y w_y = c x^k y^s + d \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*x^n*D[w[x, y], x] + b*x^m*y*D[w[x, y], y] == c*x^k*y^s + d; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to x^{1-n} \left (\frac {d}{a-a n}-\frac {c x^k y^s e^{-\frac {b s x^{m-n+1}}{a m-a n+a}} \left (-\frac {b s x^{m-n+1}}{a m-a n+a}\right )^{\frac {-k+n-1}{m-n+1}} \text {Gamma}\left (\frac {k-n+1}{m-n+1},-\frac {b s x^{m-n+1}}{a m-a n+a}\right )}{a (m-n+1)}\right )+c_1\left (y e^{-\frac {b x^{m-n+1}}{a m-a n+a}}\right )\right \}\right \}\]
Maple ✓
restart; pde :=a*x^n*diff(w(x,y),x) + n*x^m*y*diff(w(x,y),y) =c*x^k*y^s+d; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[w \left ( x,y \right ) =\int ^{x}\!{\frac {1}{a} \left ( {{\it \_a}}^{k-n}c \left ( y{{\rm e}^{-{\frac {n \left ( {x}^{-n+m+1}-{{\it \_a}}^{-n+m+1} \right ) }{a \left ( -n+m+1 \right ) }}}} \right ) ^{s}+{{\it \_a}}^{-n}d \right ) }{d{\it \_a}}+{\it \_F1} \left ( y{{\rm e}^{-{\frac {{x}^{-n+m+1}n}{a \left ( -n+m+1 \right ) }}}} \right ) \] Result has unresolved integral
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Added Feb. 9, 2019.
Problem Chapter 3.2.4.9 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ a x^n w_x +(b x^m y + c x^k) w_y = s x^p y^q + d \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*x^n*D[w[x, y], x] + (b*x^m*y + c*x^k)*D[w[x, y], y] == s*x^p*y^q + d; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \int _1^x\frac {K[1]^{-n} \left (s \left (b^{-\frac {k+1}{m-n+1}} \exp \left (-\frac {b \left (x^{m-n+1}-K[1]^{m-n+1}\right )}{a (m-n+1)}\right ) (a (m-n+1))^{-\frac {m}{m-n+1}} \left (b^{\frac {n}{m-n+1}} c e^{\frac {b x^{m-n+1}}{m a-n a+a}} \text {Gamma}\left (\frac {k-n+1}{m-n+1},\frac {b x^{m-n+1}}{m a-n a+a}\right ) (a (m-n+1))^{\frac {k}{m-n+1}}-b^{\frac {n}{m-n+1}} c e^{\frac {b x^{m-n+1}}{m a-n a+a}} \text {Gamma}\left (\frac {k-n+1}{m-n+1},\frac {b K[1]^{m-n+1}}{m a-n a+a}\right ) (a (m-n+1))^{\frac {k}{m-n+1}}+b^{\frac {k+1}{m-n+1}} y (a (m-n+1))^{\frac {m}{m-n+1}}\right )\right )^q K[1]^p+d\right )}{a}dK[1]+c_1\left (c (a (m-n+1))^{\frac {k-m}{m-n+1}} b^{\frac {-k+n-1}{m-n+1}} \text {Gamma}\left (\frac {k-n+1}{m-n+1},\frac {b x^{m-n+1}}{a m-a n+a}\right )+y e^{-\frac {b x^{m-n+1}}{a m-a n+a}}\right )\right \}\right \}\]
Maple ✓
restart; pde :=a*x^n*diff(w(x,y),x) + n*x^m*y*diff(w(x,y),y) =s*x^p*y^q+d; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[w \left ( x,y \right ) =\int ^{x}\!{\frac {1}{a} \left ( {{\it \_a}}^{-n+p}s \left ( y{{\rm e}^{-{\frac {n \left ( {x}^{-n+m+1}-{{\it \_a}}^{-n+m+1} \right ) }{a \left ( -n+m+1 \right ) }}}} \right ) ^{q}+{{\it \_a}}^{-n}d \right ) }{d{\it \_a}}+{\it \_F1} \left ( y{{\rm e}^{-{\frac {{x}^{-n+m+1}n}{a \left ( -n+m+1 \right ) }}}} \right ) \]
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Added Feb. 9, 2019.
Problem Chapter 3.2.4.10 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ a x^n w_x +(b x^m y^k + c x^r y) w_y = s x^p y^q + d \]
Mathematica ✗
ClearAll["Global`*"]; pde = a*x^n*D[w[x, y], x] + (b*x^m*y^k + c*x^r*y)*D[w[x, y], y] == s*x^p*y^q + d; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
Failed
Maple ✓
restart; pde :=a*x^n*diff(w(x,y),x) +(b*x^m*y^k + c*x^r*y)*diff(w(x,y),y) =s*x^p*y^q+d; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[w \left ( x,y \right ) =\int ^{x}\!{\frac {1}{a} \left ( {{\it \_a}}^{-n+p}s \left ( \left ( {\frac {1}{a \left ( -n+m+1 \right ) \left ( m-2\,n+r+2 \right ) \left ( m-3\,n+2\,r+3 \right ) c{y}^{ \left ( n-r-1 \right ) ^{-1}}} \left ( -a{{\it \_a}}^{-r+m}{y}^{ \left ( n-r-1 \right ) ^{-1}}{y}^{{\frac {r}{n-r-1}}}{y}^{{\frac {kn}{n-r-1}}}b{{\rm e}^{{\frac {c{x}^{-n+r+1}k}{ \left ( n-r-1 \right ) a}}}} \left ( {\frac {c \left ( k-1 \right ) }{ \left ( n-r-1 \right ) a}} \right ) ^{{\frac {-n+m+1}{n-r-1}}}{{\rm e}^{-1/2\,{\frac {{{\it \_a}}^{-n+r+1}c \left ( k-1 \right ) }{ \left ( n-r-1 \right ) a}}}} \left ( {\frac {c \left ( k-1 \right ) }{ \left ( n-r-1 \right ) a}} \right ) ^{{\frac {n-m-1}{n-r-1}}} \left ( {\frac {{{\it \_a}}^{-n+r+1}c \left ( k-1 \right ) }{ \left ( n-r-1 \right ) a}} \right ) ^{{\frac {m-2\,n+r+2}{2\,n-2\,r-2}}} \left ( n-r-1 \right ) \left ( m-2\,n+r+2 \right ) ^{2} \WhittakerM \left ( {\frac {2\,n-r-2-m}{2\,n-2\,r-2}},{\frac {-m+3\,n-2\,r-3}{2\,n-2\,r-2}},{\frac {{{\it \_a}}^{-n+r+1}c \left ( k-1 \right ) }{ \left ( n-r-1 \right ) a}} \right ) + \left ( {\frac {c \left ( k-1 \right ) }{ \left ( n-r-1 \right ) a}} \right ) ^{{\frac {-n+m+1}{n-r-1}}} \left ( -{{\it \_a}}^{-n+r+1}c \left ( k-1 \right ) +a \left ( m-2\,n+r+2 \right ) \right ) {y}^{ \left ( n-r-1 \right ) ^{-1}}b{{\rm e}^{{\frac {c{x}^{-n+r+1}k}{ \left ( n-r-1 \right ) a}}}}{{\it \_a}}^{-r+m}{{\rm e}^{-1/2\,{\frac {{{\it \_a}}^{-n+r+1}c \left ( k-1 \right ) }{ \left ( n-r-1 \right ) a}}}} \left ( {\frac {c \left ( k-1 \right ) }{ \left ( n-r-1 \right ) a}} \right ) ^{{\frac {n-m-1}{n-r-1}}}{y}^{{\frac {r}{n-r-1}}}{y}^{{\frac {kn}{n-r-1}}} \left ( {\frac {{{\it \_a}}^{-n+r+1}c \left ( k-1 \right ) }{ \left ( n-r-1 \right ) a}} \right ) ^{{\frac {m-2\,n+r+2}{2\,n-2\,r-2}}} \left ( n-r-1 \right ) ^{2} \WhittakerM \left ( {\frac {-m+r}{2\,n-2\,r-2}},{\frac {-m+3\,n-2\,r-3}{2\,n-2\,r-2}},{\frac {{{\it \_a}}^{-n+r+1}c \left ( k-1 \right ) }{ \left ( n-r-1 \right ) a}} \right ) + \left ( m-2\,n+r+2 \right ) \left ( {y}^{ \left ( n-r-1 \right ) ^{-1}}{y}^{{\frac {r}{n-r-1}}}{y}^{{\frac {kn}{n-r-1}}}b{{\rm e}^{{\frac {c{x}^{-n+r+1}k}{ \left ( n-r-1 \right ) a}}}} \left ( k-1 \right ) \int \!{{\rm e}^{-{\frac {{x}^{-n+r+1}c \left ( k-1 \right ) }{ \left ( n-r-1 \right ) a}}}}{x}^{-n+m}\,{\rm d}x+{y}^{{\frac {rk}{n-r-1}}}{y}^{{\frac {k}{n-r-1}}}{y}^{{\frac {n}{n-r-1}}}{{\rm e}^{{\frac {c{x}^{-n+r+1}}{ \left ( n-r-1 \right ) a}}}}a \right ) c \left ( m-3\,n+2\,r+3 \right ) \left ( -n+m+1 \right ) \right ) \left ( {y}^{{\frac {kn}{n-r-1}}} \right ) ^{-1} \left ( {{\rm e}^{{\frac {c{x}^{-n+r+1}k}{ \left ( n-r-1 \right ) a}}}} \right ) ^{-1} \left ( {y}^{{\frac {r}{n-r-1}}} \right ) ^{-1}} \right ) ^{- \left ( k-1 \right ) ^{-1}}{{\rm e}^{-{\frac {{{\it \_a}}^{-n+r+1}c}{ \left ( n-r-1 \right ) a}}}} \right ) ^{q}+{{\it \_a}}^{-n}d \right ) }{d{\it \_a}}+{\it \_F1} \left ( {\frac {1}{a \left ( -n+m+1 \right ) \left ( m-2\,n+r+2 \right ) \left ( m-3\,n+2\,r+3 \right ) c{y}^{ \left ( n-r-1 \right ) ^{-1}}} \left ( a{x}^{-r+m}{y}^{ \left ( n-r-1 \right ) ^{-1}}{y}^{{\frac {r}{n-r-1}}}{y}^{{\frac {kn}{n-r-1}}}{{\rm e}^{{\frac {c{x}^{-n+r+1}k}{ \left ( n-r-1 \right ) a}}}}b \left ( {\frac {c \left ( k-1 \right ) }{ \left ( n-r-1 \right ) a}} \right ) ^{{\frac {-n+m+1}{n-r-1}}}{{\rm e}^{-1/2\,{\frac {{x}^{-n+r+1}c \left ( k-1 \right ) }{ \left ( n-r-1 \right ) a}}}} \left ( {\frac {c \left ( k-1 \right ) }{ \left ( n-r-1 \right ) a}} \right ) ^{{\frac {n-m-1}{n-r-1}}} \left ( {\frac {{x}^{-n+r+1}c \left ( k-1 \right ) }{ \left ( n-r-1 \right ) a}} \right ) ^{{\frac {m-2\,n+r+2}{2\,n-2\,r-2}}} \left ( n-r-1 \right ) \left ( m-2\,n+r+2 \right ) ^{2} \WhittakerM \left ( {\frac {2\,n-r-2-m}{2\,n-2\,r-2}},{\frac {-m+3\,n-2\,r-3}{2\,n-2\,r-2}},{\frac {{x}^{-n+r+1}c \left ( k-1 \right ) }{ \left ( n-r-1 \right ) a}} \right ) - \left ( {\frac {c \left ( k-1 \right ) }{ \left ( n-r-1 \right ) a}} \right ) ^{{\frac {-n+m+1}{n-r-1}}}b \left ( {\frac {c \left ( k-1 \right ) }{ \left ( n-r-1 \right ) a}} \right ) ^{{\frac {n-m-1}{n-r-1}}}{x}^{-r+m}{y}^{ \left ( n-r-1 \right ) ^{-1}} \left ( {\frac {{x}^{-n+r+1}c \left ( k-1 \right ) }{ \left ( n-r-1 \right ) a}} \right ) ^{{\frac {m-2\,n+r+2}{2\,n-2\,r-2}}} \left ( -{x}^{-n+r+1}c \left ( k-1 \right ) +a \left ( m-2\,n+r+2 \right ) \right ) {{\rm e}^{-1/2\,{\frac {{x}^{-n+r+1}c \left ( k-1 \right ) }{ \left ( n-r-1 \right ) a}}}}{y}^{{\frac {kn}{n-r-1}}}{{\rm e}^{{\frac {c{x}^{-n+r+1}k}{ \left ( n-r-1 \right ) a}}}}{y}^{{\frac {r}{n-r-1}}} \left ( n-r-1 \right ) ^{2} \WhittakerM \left ( {\frac {-m+r}{2\,n-2\,r-2}},{\frac {-m+3\,n-2\,r-3}{2\,n-2\,r-2}},{\frac {{x}^{-n+r+1}c \left ( k-1 \right ) }{ \left ( n-r-1 \right ) a}} \right ) +{y}^{{\frac {rk}{n-r-1}}}{y}^{{\frac {k}{n-r-1}}}{y}^{{\frac {n}{n-r-1}}}{{\rm e}^{{\frac {c{x}^{-n+r+1}}{ \left ( n-r-1 \right ) a}}}}ac \left ( -n+m+1 \right ) \left ( m-2\,n+r+2 \right ) \left ( m-3\,n+2\,r+3 \right ) \right ) \left ( {y}^{{\frac {kn}{n-r-1}}} \right ) ^{-1} \left ( {{\rm e}^{{\frac {c{x}^{-n+r+1}k}{ \left ( n-r-1 \right ) a}}}} \right ) ^{-1} \left ( {y}^{{\frac {r}{n-r-1}}} \right ) ^{-1}} \right ) \]
____________________________________________________________________________________
Added Feb. 9, 2019.
Problem Chapter 3.2.4.11 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ a y^k w_x + b x^m w_y = c x^m + d \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*y^k*D[w[x, y], x] + b*x^m*D[w[x, y], y] == c*x^m + d; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (\frac {y^{k+1}}{k+1}-\frac {b x^{m+1}}{a m+a}\right )+\frac {\left (\left (y^{-k-1}\right )^{-\frac {1}{k+1}}\right )^{-k} \left (b d x \left (\frac {a (m+1) y^{k+1}}{a (m+1) y^{k+1}-b (k+1) x^{m+1}}\right )^{\frac {k}{k+1}} \, _2F_1\left (\frac {k}{k+1},\frac {1}{m+1};1+\frac {1}{m+1};\frac {b (k+1) x^{m+1}}{b (k+1) x^{m+1}-a (m+1) y^{k+1}}\right )+a c y^{k+1}\right )}{a b}\right \}\right \}\]
Maple ✓
restart; pde :=a*y^k*diff(w(x,y),x) +b*x^n*diff(w(x,y),y) =c*x^m+d; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[w \left ( x,y \right ) =\int ^{x}\!{\frac {c{{\it \_a}}^{m}+d}{a} \left ( \left ( {\frac {-{x}^{n+1}b \left ( k+1 \right ) +{y}^{k+1}a \left ( n+1 \right ) +{{\it \_a}}^{n+1}b \left ( k+1 \right ) }{a \left ( n+1 \right ) }} \right ) ^{ \left ( k+1 \right ) ^{-1}} \right ) ^{-k}}{d{\it \_a}}+{\it \_F1} \left ( {\frac {-{x}^{n+1}b \left ( k+1 \right ) +{y}^{k+1}a \left ( n+1 \right ) }{a \left ( n+1 \right ) }} \right ) \]
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