____________________________________________________________________________________
Added Feb. 9, 2019.
Problem Chapter 3.3.2.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b w_y = c y e^{\lambda x} + k x e^{\mu y} \]
Mathematica ✓
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B, v, f, h, q, p, delta]; pde = a*D[w[x, y], x] + b*D[w[x, y], y] == c*y*Exp[lambda*x] + k*x*Exp[mu*y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to -\frac {a^3 k \lambda ^2 e^{\frac {\mu (a y-b x)}{a}+\frac {b \mu x}{a}}-a^2 b^2 \lambda ^2 \mu ^2 c_1\left (\frac {a y-b x}{a}\right )-a^2 b k \lambda ^2 \mu x e^{\frac {\mu (a y-b x)}{a}+\frac {b \mu x}{a}}-a b^2 c \lambda \mu ^2 y e^{\lambda x}+b^3 c \mu ^2 e^{\lambda x}}{a^2 b^2 \lambda ^2 \mu ^2}\right \}\right \} \]
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';f:='f'; C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v';q:='q';p:='p';l:='l'; pde :=a*diff(w(x,y),x) +b*diff(w(x,y),y) =c*y*exp(lambda*x)+k*x*exp(mu*y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[ w \left ( x,y \right ) ={\frac {kx}{b\mu }{{\rm e}^{{\frac { \left ( ya-bx \right ) \mu }{a}}+{\frac {b\mu \,x}{a}}}}}+{\frac {cy{{\rm e}^{\lambda \,x}}}{a\lambda }}+{\frac {1}{{b}^{2}{\mu }^{2}{a}^{2}{\lambda }^{2}} \left ( {\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) {b}^{2}{\mu }^{2}{a}^{2}{\lambda }^{2}-{{\rm e}^{\lambda \,x}}c{b}^{3}{\mu }^{2}-k{a}^{3}{{\rm e}^{{\frac { \left ( ya-bx \right ) \mu }{a}}+{\frac {b\mu \,x}{a}}}}{\lambda }^{2} \right ) } \]
____________________________________________________________________________________
Added Feb. 9, 2019.
Problem Chapter 3.3.2.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + a w_y = a x^k e^{\lambda y} \]
Mathematica ✓
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B, v, f, h, q, p, delta]; pde = D[w[x, y], x] + a*D[w[x, y], y] == a*x^k*Exp[lambda*y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to \frac {(-a \lambda x)^{-k} \left (x^k e^{\lambda (y-a x)} \text {Gamma}(k+1,-a \lambda x)+\lambda c_1(y-a x) (-a \lambda x)^k\right )}{\lambda }\right \}\right \} \]
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';f:='f'; C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v';q:='q';p:='p';l:='l'; pde := diff(w(x,y),x) +a*diff(w(x,y),y) =a*x^k*exp(lambda*y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[ w \left ( x,y \right ) =-{\frac { \left ( -ax\lambda \right ) ^{-k}k\Gamma \left ( k \right ) {x}^{k}{{\rm e}^{ \left ( -ax+y \right ) \lambda }}- \left ( -ax\lambda \right ) ^{-k}\Gamma \left ( k,-ax\lambda \right ) k{x}^{k}{{\rm e}^{ \left ( -ax+y \right ) \lambda }}-{x}^{k}{{\rm e}^{ax\lambda + \left ( -ax+y \right ) \lambda }}-{\it \_F1} \left ( -ax+y \right ) \lambda }{\lambda }} \]
____________________________________________________________________________________
Added Feb. 9, 2019.
Problem Chapter 3.3.2.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + (a y+b e^{\lambda x}) w_y = c e^{\beta x} \]
Mathematica ✓
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B, v, f, h, q, p, delta]; pde = D[w[x, y], x] + (a*y + b*Exp[lambda*x])*D[w[x, y], y] == c*Exp[beta*x]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to \frac {\beta c_1\left (\frac {e^{-a x} \left (a y+b e^{\lambda x}-\lambda y\right )}{a-\lambda }\right )+c e^{\beta x}}{\beta }\right \}\right \} \]
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';f:='f'; C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v';q:='q';p:='p';l:='l'; pde := diff(w(x,y),x) +(a*y+b*exp(lambda*x))*diff(w(x,y),y) =c*exp(beta*x); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[ w \left ( x,y \right ) ={\frac {1}{\beta } \left ( {\it \_F1} \left ( {\frac { \left ( y{{\rm e}^{x \left ( a-\lambda \right ) }}a-y{{\rm e}^{x \left ( a-\lambda \right ) }}\lambda +{{\rm e}^{ax}}b \right ) {{\rm e}^{-x \left ( 2\,a-\lambda \right ) }}}{a-\lambda }} \right ) \beta +c{{\rm e}^{\beta \,x}} \right ) } \]
____________________________________________________________________________________
Added Feb. 9, 2019.
Problem Chapter 3.3.2.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + (a y e^{\lambda x}+b e^{\beta x} y^k) w_y = c e^{\mu x} \]
Mathematica ✓
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B, v, f, h, q, p, delta]; pde = D[w[x, y], x] + (a*y*Exp[lambda*x] + b*Exp[beta*x]*y^k)*D[w[x, y], y] == c*Exp[mu*x]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to \frac {\mu c_1\left (y^{-k} e^{-\frac {a e^{\lambda x}}{\lambda }} \left (y^k \left (-e^{\frac {a e^{\lambda x}}{\lambda }}\right ) \left (\int _1^x b e^{\beta K[1]-\frac {a (1-k) e^{\lambda K[1]}}{\lambda }} \, dK[1]\right )+k y^k e^{\frac {a e^{\lambda x}}{\lambda }} \left (\int _1^x b e^{\beta K[1]-\frac {a (1-k) e^{\lambda K[1]}}{\lambda }} \, dK[1]\right )+y e^{\frac {a k e^{\lambda x}}{\lambda }}\right )\right )+c e^{\mu x}}{\mu }\right \}\right \} \]
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';f:='f'; C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v';q:='q';p:='p';l:='l'; pde := diff(w(x,y),x) +(a*y*exp(lambda*x)+b*exp(beta*x)*y^k)*diff(w(x,y),y) =c*exp(mu*x); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[ w \left ( x,y \right ) ={\frac {1}{\mu } \left ( {\it \_F1} \left ( {\frac {y}{{y}^{k}}{{\rm e}^{{\frac {{{\rm e}^{\lambda \,x}}ak}{\lambda }}}} \left ( {{\rm e}^{{\frac {a{{\rm e}^{\lambda \,x}}}{\lambda }}}} \right ) ^{-1}}+kb\int \!{{\rm e}^{{\frac {{{\rm e}^{\lambda \,x}}ak+\beta \,x\lambda -a{{\rm e}^{\lambda \,x}}}{\lambda }}}}\,{\rm d}x-b\int \!{{\rm e}^{{\frac {{{\rm e}^{\lambda \,x}}ak+\beta \,x\lambda -a{{\rm e}^{\lambda \,x}}}{\lambda }}}}\,{\rm d}x \right ) \mu +c{{\rm e}^{\mu \,x}} \right ) } \]
____________________________________________________________________________________
Added Feb. 9, 2019.
Problem Chapter 3.3.2.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + (a x^k+b x^n e^{\lambda y}) w_y = c e^{\beta x} \]
Mathematica ✓
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B, v, f, h, q, p, delta]; pde = D[w[x, y], x] + (a*x^k + b*x^n*Exp[lambda*y])*D[w[x, y], y] == c*Exp[beta*x]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to \frac {\beta c_1\left (\frac {b \lambda x^{n+1} \left (-\frac {a \lambda x^{k+1}}{k+1}\right )^{-\frac {n}{k+1}-\frac {1}{k+1}} \text {Gamma}\left (\frac {n+1}{k+1},-\frac {a \lambda x^{k+1}}{k+1}\right )-e^{-\frac {\lambda \left (-a x^{k+1}+k y+y\right )}{k+1}}-k e^{-\frac {\lambda \left (-a x^{k+1}+k y+y\right )}{k+1}}}{a b k^2 \lambda ^2-a b k \lambda ^2 n+a b k \lambda ^2-a b \lambda ^2 n}\right )+c e^{\beta x}}{\beta }\right \}\right \} \]
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';f:='f'; C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v';q:='q';p:='p';l:='l'; pde := diff(w(x,y),x) +(a*x^k+b*x^n*exp(lambda*y))*diff(w(x,y),y) =c*exp(beta*x); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[ w \left ( x,y \right ) ={\frac {1}{\beta } \left ( {\it \_F1} \left ( -{\frac {1}{a\lambda \, \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 ( 2\, \left ( -{\frac {{x}^{k+1}\lambda \,a}{k+1}} \right ) ^{-1/2\,{\frac {k+n+2}{k+1}}}{{\rm e}^{1/2\,{\frac {{x}^{k+1}\lambda \,a}{k+1}}}} \WhittakerM \left ( -1/2\,{\frac {k-n}{k+1}},1/2\,{\frac {2\,k+n+3}{k+1}},-{\frac {{x}^{k+1}\lambda \,a}{k+1}} \right ) abk\lambda \,{x}^{n+1}+ \left ( -{\frac {{x}^{k+1}\lambda \,a}{k+1}} \right ) ^{-1/2\,{\frac {k+n+2}{k+1}}}{{\rm e}^{1/2\,{\frac {{x}^{k+1}\lambda \,a}{k+1}}}} \WhittakerM \left ( -1/2\,{\frac {k-n}{k+1}},1/2\,{\frac {2\,k+n+3}{k+1}},-{\frac {{x}^{k+1}\lambda \,a}{k+1}} \right ) ab{k}^{2}\lambda \,{x}^{n+1}+6\,{{\rm e}^{{\frac {\lambda \, \left ( {x}^{k+1}a-ky-y \right ) }{k+1}}}}a{n}^{2}+2\,{{\rm e}^{{\frac {\lambda \, \left ( {x}^{k+1}a-ky-y \right ) }{k+1}}}}a{k}^{2}+{{\rm e}^{{\frac {\lambda \, \left ( {x}^{k+1}a-ky-y \right ) }{k+1}}}}a{n}^{3}+7\,{{\rm e}^{{\frac {\lambda \, \left ( {x}^{k+1}a-ky-y \right ) }{k+1}}}}ka+11\,{{\rm e}^{{\frac {\lambda \, \left ( {x}^{k+1}a-ky-y \right ) }{k+1}}}}an-4\,{x}^{-k+n} \left ( -{\frac {{x}^{k+1}\lambda \,a}{k+1}} \right ) ^{-1/2\,{\frac {k+n+2}{k+1}}}{{\rm e}^{1/2\,{\frac {{x}^{k+1}\lambda \,a}{k+1}}}} \WhittakerM \left ( 1/2\,{\frac {k+n+2}{k+1}},1/2\,{\frac {2\,k+n+3}{k+1}},-{\frac {{x}^{k+1}\lambda \,a}{k+1}} \right ) b-5\,{x}^{-k+n} \left ( -{\frac {{x}^{k+1}\lambda \,a}{k+1}} \right ) ^{-1/2\,{\frac {k+n+2}{k+1}}}{{\rm e}^{1/2\,{\frac {{x}^{k+1}\lambda \,a}{k+1}}}} \WhittakerM \left ( -1/2\,{\frac {k-n}{k+1}},1/2\,{\frac {2\,k+n+3}{k+1}},-{\frac {{x}^{k+1}\lambda \,a}{k+1}} \right ) bk-4\,{x}^{-k+n} \left ( -{\frac {{x}^{k+1}\lambda \,a}{k+1}} \right ) ^{-1/2\,{\frac {k+n+2}{k+1}}}{{\rm e}^{1/2\,{\frac {{x}^{k+1}\lambda \,a}{k+1}}}} \WhittakerM \left ( -1/2\,{\frac {k-n}{k+1}},1/2\,{\frac {2\,k+n+3}{k+1}},-{\frac {{x}^{k+1}\lambda \,a}{k+1}} \right ) b{k}^{2}-{x}^{-k+n} \left ( -{\frac {{x}^{k+1}\lambda \,a}{k+1}} \right ) ^{-1/2\,{\frac {k+n+2}{k+1}}}{{\rm e}^{1/2\,{\frac {{x}^{k+1}\lambda \,a}{k+1}}}} \WhittakerM \left ( -1/2\,{\frac {k-n}{k+1}},1/2\,{\frac {2\,k+n+3}{k+1}},-{\frac {{x}^{k+1}\lambda \,a}{k+1}} \right ) b{k}^{3}-{x}^{-k+n} \left ( -{\frac {{x}^{k+1}\lambda \,a}{k+1}} \right ) ^{-1/2\,{\frac {k+n+2}{k+1}}}{{\rm e}^{1/2\,{\frac {{x}^{k+1}\lambda \,a}{k+1}}}} \WhittakerM \left ( -1/2\,{\frac {k-n}{k+1}},1/2\,{\frac {2\,k+n+3}{k+1}},-{\frac {{x}^{k+1}\lambda \,a}{k+1}} \right ) bn-8\,{x}^{-k+n} \left ( -{\frac {{x}^{k+1}\lambda \,a}{k+1}} \right ) ^{-1/2\,{\frac {k+n+2}{k+1}}}{{\rm e}^{1/2\,{\frac {{x}^{k+1}\lambda \,a}{k+1}}}} \WhittakerM \left ( 1/2\,{\frac {k+n+2}{k+1}},1/2\,{\frac {2\,k+n+3}{k+1}},-{\frac {{x}^{k+1}\lambda \,a}{k+1}} \right ) bk-5\,{x}^{-k+n} \left ( -{\frac {{x}^{k+1}\lambda \,a}{k+1}} \right ) ^{-1/2\,{\frac {k+n+2}{k+1}}}{{\rm e}^{1/2\,{\frac {{x}^{k+1}\lambda \,a}{k+1}}}} \WhittakerM \left ( 1/2\,{\frac {k+n+2}{k+1}},1/2\,{\frac {2\,k+n+3}{k+1}},-{\frac {{x}^{k+1}\lambda \,a}{k+1}} \right ) b{k}^{2}-{x}^{-k+n} \left ( -{\frac {{x}^{k+1}\lambda \,a}{k+1}} \right ) ^{-1/2\,{\frac {k+n+2}{k+1}}}{{\rm e}^{1/2\,{\frac {{x}^{k+1}\lambda \,a}{k+1}}}} \WhittakerM \left ( 1/2\,{\frac {k+n+2}{k+1}},1/2\,{\frac {2\,k+n+3}{k+1}},-{\frac {{x}^{k+1}\lambda \,a}{k+1}} \right ) b{k}^{3}-4\,{x}^{-k+n} \left ( -{\frac {{x}^{k+1}\lambda \,a}{k+1}} \right ) ^{-1/2\,{\frac {k+n+2}{k+1}}}{{\rm e}^{1/2\,{\frac {{x}^{k+1}\lambda \,a}{k+1}}}} \WhittakerM \left ( 1/2\,{\frac {k+n+2}{k+1}},1/2\,{\frac {2\,k+n+3}{k+1}},-{\frac {{x}^{k+1}\lambda \,a}{k+1}} \right ) bn-{x}^{-k+n} \left ( -{\frac {{x}^{k+1}\lambda \,a}{k+1}} \right ) ^{-1/2\,{\frac {k+n+2}{k+1}}}{{\rm e}^{1/2\,{\frac {{x}^{k+1}\lambda \,a}{k+1}}}} \WhittakerM \left ( 1/2\,{\frac {k+n+2}{k+1}},1/2\,{\frac {2\,k+n+3}{k+1}},-{\frac {{x}^{k+1}\lambda \,a}{k+1}} \right ) b{n}^{2}-2\,{x}^{-k+n} \left ( -{\frac {{x}^{k+1}\lambda \,a}{k+1}} \right ) ^{-1/2\,{\frac {k+n+2}{k+1}}}{{\rm e}^{1/2\,{\frac {{x}^{k+1}\lambda \,a}{k+1}}}} \WhittakerM \left ( -1/2\,{\frac {k-n}{k+1}},1/2\,{\frac {2\,k+n+3}{k+1}},-{\frac {{x}^{k+1}\lambda \,a}{k+1}} \right ) b+2\,{{\rm e}^{{\frac {\lambda \, \left ( {x}^{k+1}a-ky-y \right ) }{k+1}}}}a{k}^{2}n+3\,{{\rm e}^{{\frac {\lambda \, \left ( {x}^{k+1}a-ky-y \right ) }{k+1}}}}ak{n}^{2}+10\,{{\rm e}^{{\frac {\lambda \, \left ( {x}^{k+1}a-ky-y \right ) }{k+1}}}}kan-2\,{x}^{-k+n} \left ( -{\frac {{x}^{k+1}\lambda \,a}{k+1}} \right ) ^{-1/2\,{\frac {k+n+2}{k+1}}}{{\rm e}^{1/2\,{\frac {{x}^{k+1}\lambda \,a}{k+1}}}} \WhittakerM \left ( -1/2\,{\frac {k-n}{k+1}},1/2\,{\frac {2\,k+n+3}{k+1}},-{\frac {{x}^{k+1}\lambda \,a}{k+1}} \right ) bkn-{x}^{-k+n} \left ( -{\frac {{x}^{k+1}\lambda \,a}{k+1}} \right ) ^{-1/2\,{\frac {k+n+2}{k+1}}}{{\rm e}^{1/2\,{\frac {{x}^{k+1}\lambda \,a}{k+1}}}} \WhittakerM \left ( -1/2\,{\frac {k-n}{k+1}},1/2\,{\frac {2\,k+n+3}{k+1}},-{\frac {{x}^{k+1}\lambda \,a}{k+1}} \right ) b{k}^{2}n-6\,{x}^{-k+n} \left ( -{\frac {{x}^{k+1}\lambda \,a}{k+1}} \right ) ^{-1/2\,{\frac {k+n+2}{k+1}}}{{\rm e}^{1/2\,{\frac {{x}^{k+1}\lambda \,a}{k+1}}}} \WhittakerM \left ( 1/2\,{\frac {k+n+2}{k+1}},1/2\,{\frac {2\,k+n+3}{k+1}},-{\frac {{x}^{k+1}\lambda \,a}{k+1}} \right ) bkn-{x}^{-k+n} \left ( -{\frac {{x}^{k+1}\lambda \,a}{k+1}} \right ) ^{-1/2\,{\frac {k+n+2}{k+1}}}{{\rm e}^{1/2\,{\frac {{x}^{k+1}\lambda \,a}{k+1}}}} \WhittakerM \left ( 1/2\,{\frac {k+n+2}{k+1}},1/2\,{\frac {2\,k+n+3}{k+1}},-{\frac {{x}^{k+1}\lambda \,a}{k+1}} \right ) bk{n}^{2}-2\,{x}^{-k+n} \left ( -{\frac {{x}^{k+1}\lambda \,a}{k+1}} \right ) ^{-1/2\,{\frac {k+n+2}{k+1}}}{{\rm e}^{1/2\,{\frac {{x}^{k+1}\lambda \,a}{k+1}}}} \WhittakerM \left ( 1/2\,{\frac {k+n+2}{k+1}},1/2\,{\frac {2\,k+n+3}{k+1}},-{\frac {{x}^{k+1}\lambda \,a}{k+1}} \right ) b{k}^{2}n+ \left ( -{\frac {{x}^{k+1}\lambda \,a}{k+1}} \right ) ^{-1/2\,{\frac {k+n+2}{k+1}}}{{\rm e}^{1/2\,{\frac {{x}^{k+1}\lambda \,a}{k+1}}}} \WhittakerM \left ( -1/2\,{\frac {k-n}{k+1}},1/2\,{\frac {2\,k+n+3}{k+1}},-{\frac {{x}^{k+1}\lambda \,a}{k+1}} \right ) ab\lambda \,{x}^{n+1}+6\,{{\rm e}^{{\frac {\lambda \, \left ( {x}^{k+1}a-ky-y \right ) }{k+1}}}}a \right ) } \right ) \beta +c{{\rm e}^{\beta \,x}} \right ) } \]
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Added Feb. 9, 2019.
Problem Chapter 3.3.2.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ x w_x + y w_y = a x e^{\lambda x+ \mu y} \]
Mathematica ✓
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B, v, f, h, q, p, delta]; pde = x*D[w[x, y], x] + y*D[w[x, y], y] == a*x*Exp[lambda*x + mu*y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to \frac {a x e^{x \left (\lambda +\frac {\mu y}{x}\right )}+\lambda x c_1\left (\frac {y}{x}\right )+\mu y c_1\left (\frac {y}{x}\right )}{\lambda x+\mu y}\right \}\right \} \]
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';f:='f'; C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v';q:='q';p:='p';l:='l'; pde := x*diff(w(x,y),x) +y*diff(w(x,y),y) =a*x*exp(lambda *x+ mu* y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[ w \left ( x,y \right ) ={a{{\rm e}^{\lambda \,x+\mu \,y}} \left ( {\frac {\mu \,y}{x}}+\lambda \right ) ^{-1}}+{\it \_F1} \left ( {\frac {y}{x}} \right ) \]
____________________________________________________________________________________
Added Feb. 9, 2019.
Problem Chapter 3.3.2.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ x w_x + y w_y = a y e^{\lambda x} + b x e^{\mu y} \]
Mathematica ✓
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B, v, f, h, q, p, delta]; pde = x*D[w[x, y], x] + y*D[w[x, y], y] == a*y*Exp[lambda*x] + b*x*Exp[mu*y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to \frac {a \mu y^2 e^{\lambda x}+b \lambda x^2 e^{\mu y}+\lambda \mu x y c_1\left (\frac {y}{x}\right )}{\lambda \mu x y}\right \}\right \} \]
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';f:='f'; C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v';q:='q';p:='p';l:='l'; pde := x*diff(w(x,y),x) +y*diff(w(x,y),y) =a*y*exp(lambda*x) + b*x*exp(mu*y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[ w \left ( x,y \right ) ={\frac {bx{{\rm e}^{\mu \,y}}}{\mu \,y}}+{\frac {{{\rm e}^{\lambda \,x}}ay}{\lambda \,x}}+{\it \_F1} \left ( {\frac {y}{x}} \right ) \]
____________________________________________________________________________________
Added Feb. 9, 2019.
Problem Chapter 3.3.2.8 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a x^k w_x + b e^{\lambda y} w_y = c x^n+s \]
Mathematica ✓
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B, v, f, h, q, p, delta]; pde = a*x^k*D[w[x, y], x] + b*Exp[lambda*y]*D[w[x, y], y] == c*x^n + s; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to \frac {x^{-k} \left (a k^2 x^k c_1\left (\frac {x^{-k} e^{-\lambda y} \left (a x^k-a k x^k+b \lambda x e^{\lambda y}\right )}{a (k-1) \lambda }\right )+a n x^k c_1\left (\frac {x^{-k} e^{-\lambda y} \left (a x^k-a k x^k+b \lambda x e^{\lambda y}\right )}{a (k-1) \lambda }\right )-a k n x^k c_1\left (\frac {x^{-k} e^{-\lambda y} \left (a x^k-a k x^k+b \lambda x e^{\lambda y}\right )}{a (k-1) \lambda }\right )+a x^k c_1\left (\frac {x^{-k} e^{-\lambda y} \left (a x^k-a k x^k+b \lambda x e^{\lambda y}\right )}{a (k-1) \lambda }\right )-2 a k x^k c_1\left (\frac {x^{-k} e^{-\lambda y} \left (a x^k-a k x^k+b \lambda x e^{\lambda y}\right )}{a (k-1) \lambda }\right )-c k x^{n+1}+c x^{n+1}-k s x+n s x+s x\right )}{a (k-1) (k-n-1)}\right \}\right \} \]
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';f:='f'; C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v';q:='q';p:='p';l:='l'; pde :=a*x^k*diff(w(x,y),x) +b*exp(lambda*y)*diff(w(x,y),y) =c*x^n+s; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[ w \left ( x,y \right ) ={\frac {{x}^{1-k}s}{a \left ( 1-k \right ) }}+{\frac {{x}^{1-k+n}c}{a \left ( 1-k+n \right ) }}+{\it \_F1} \left ( {\frac {{x}^{1-k}\lambda \,b-ak{{\rm e}^{-y\lambda }}+a{{\rm e}^{-y\lambda }}}{b\lambda \, \left ( k-1 \right ) }} \right ) \]
____________________________________________________________________________________
Added Feb. 9, 2019.
Problem Chapter 3.3.2.9 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a y^k w_x + b e^{\lambda x} w_y = c e^{\mu x}+s \]
Mathematica ✓
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B, v, f, h, q, p, delta]; pde = a*y^k*D[w[x, y], x] + b*Exp[lambda*x]*D[w[x, y], y] == c*Exp[mu*x] + s; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to \frac {\left (\left (\frac {(k+1) \left (\frac {a \lambda y^{k+1}-b k e^{\lambda x}-b e^{\lambda x}}{k+1}+b e^{\lambda x}\right )}{a \lambda }\right )^{\frac {1}{k+1}}\right )^{-k} \left (c k \lambda e^{\mu x} \left (\frac {b (k+1) e^{\lambda x}}{a \lambda y^{k+1}-b k e^{\lambda x}-b e^{\lambda x}}+1\right )^{\frac {k}{k+1}} \text {Hypergeometric2F1}\left (\frac {k}{k+1},\frac {\mu }{\lambda },\frac {\lambda +\mu }{\lambda },-\frac {b (k+1) e^{\lambda x}}{a \lambda y^{k+1}-b k e^{\lambda x}-b e^{\lambda x}}\right )-k \mu s \left (\frac {e^{-\lambda x} \left (a \lambda y^{k+1}-b k e^{\lambda x}-b e^{\lambda x}\right )}{b (k+1)}+1\right )^{\frac {k}{k+1}} \text {Hypergeometric2F1}\left (\frac {k}{k+1},\frac {k}{k+1},\frac {k}{k+1}+1,-\frac {e^{-\lambda x} \left (a \lambda y^{k+1}-b k e^{\lambda x}-b e^{\lambda x}\right )}{b (k+1)}\right )-\mu s \left (\frac {e^{-\lambda x} \left (a \lambda y^{k+1}-b k e^{\lambda x}-b e^{\lambda x}\right )}{b (k+1)}+1\right )^{\frac {k}{k+1}} \text {Hypergeometric2F1}\left (\frac {k}{k+1},\frac {k}{k+1},\frac {k}{k+1}+1,-\frac {e^{-\lambda x} \left (a \lambda y^{k+1}-b k e^{\lambda x}-b e^{\lambda x}\right )}{b (k+1)}\right )+a k \lambda \mu \left (\left (\frac {(k+1) \left (\frac {a \lambda y^{k+1}-b k e^{\lambda x}-b e^{\lambda x}}{k+1}+b e^{\lambda x}\right )}{a \lambda }\right )^{\frac {1}{k+1}}\right )^k c_1\left (\frac {a \lambda y^{k+1}-b k e^{\lambda x}-b e^{\lambda x}}{a (k+1) \lambda }\right )\right )}{a k \lambda \mu }\right \}\right \} \]
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';f:='f'; C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v';q:='q';p:='p';l:='l'; pde :=a*y^k*diff(w(x,y),x) +b*exp(lambda*x)*diff(w(x,y),y) =c*exp(mu*x)+s; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[ w \left ( x,y \right ) =\int ^{x}\!{\frac {c{{\rm e}^{\mu \,{\it \_a}}}+s}{a} \left ( \left ( {\frac {{{\rm e}^{\lambda \,{\it \_a}}}bk-{{\rm e}^{\lambda \,x}}bk+{y}^{k}ya\lambda -{{\rm e}^{\lambda \,x}}b+{{\rm e}^{\lambda \,{\it \_a}}}b}{a\lambda }} \right ) ^{ \left ( k+1 \right ) ^{-1}} \right ) ^{-k}}{d{\it \_a}}+{\it \_F1} \left ( -{\frac {{{\rm e}^{\lambda \,x}}bk-{y}^{k}ya\lambda +{{\rm e}^{\lambda \,x}}b}{a\lambda }} \right ) \]
____________________________________________________________________________________
Added Feb. 9, 2019.
Problem Chapter 3.3.2.10 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a e^{\lambda x} w_x + b y^k w_y = c x^n+s \]
Mathematica ✓
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B, v, f, h, q, p, delta]; pde = a*Exp[lambda*x]*D[w[x, y], x] + b*y^k*D[w[x, y], y] == c*x^n + s; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to \frac {e^{-\lambda x} (\lambda x)^{-n} \left (-c e^{\lambda x} x^n \text {Gamma}(n+1,\lambda x)+a \lambda e^{\lambda x} (\lambda x)^n c_1\left (-\frac {y^{-k} e^{-\lambda x} \left (a \lambda y e^{\lambda x}+b y^k-b k y^k\right )}{a (k-1) \lambda }\right )-s (\lambda x)^n\right )}{a \lambda }\right \}\right \} \]
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';f:='f'; C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v';q:='q';p:='p';l:='l'; pde :=a*exp(lambda*x)*diff(w(x,y),x) +b*y^k*diff(w(x,y),y) = c*x^n+s; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[ w \left ( x,y \right ) ={\frac {{x}^{n}c \left ( \lambda \,x \right ) ^{-n/2}{{\rm e}^{-1/2\,\lambda \,x}} \WhittakerM \left ( n/2,n/2+1/2,\lambda \,x \right ) }{ \left ( n+1 \right ) a\lambda }}+{\frac {s \left ( 1-{{\rm e}^{-\lambda \,x}} \right ) }{a\lambda }}+{\it \_F1} \left ( -{\frac {{{\rm e}^{-\lambda \,x}}bk-{y}^{1-k}a\lambda -b{{\rm e}^{-\lambda \,x}}}{a\lambda }} \right ) \]
____________________________________________________________________________________
Added Feb. 9, 2019.
Problem Chapter 3.3.2.11 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a e^{\lambda y} w_x + b x^k w_y = c Exp[\mu x]+s \]
Mathematica ✗
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B, v, f, h, q, p, delta]; pde = a*Exp[lambda*y]*D[w[x, y], x] + b*x^k*D[w[x, y], y] == c*Exp[mu*x] + s; 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';f:='f'; C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v';q:='q';p:='p';l:='l'; pde :=a*exp(lambda*y)*diff(w(x,y),x) +b*x^k*diff(w(x,y),y) = c*exp(mu*x)+s; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[ w \left ( x,y \right ) =\int ^{x}\!{\frac { \left ( c{{\rm e}^{\mu \,{\it \_a}}}+s \right ) \left ( k+1 \right ) }{\lambda \,b} \left ( -{\frac { \left ( {x}^{k+1}\lambda \,b-{{\rm e}^{y\lambda }}ak-a{{\rm e}^{y\lambda }} \right ) k}{ \left ( k+1 \right ) \lambda \,b}}+{{\it \_a}}^{k+1}-{\frac {{x}^{k+1}\lambda \,b-{{\rm e}^{y\lambda }}ak-a{{\rm e}^{y\lambda }}}{ \left ( k+1 \right ) \lambda \,b}} \right ) ^{-1}}{d{\it \_a}}+{\it \_F1} \left ( -{\frac {{x}^{k+1}\lambda \,b-{{\rm e}^{y\lambda }}ak-a{{\rm e}^{y\lambda }}}{ \left ( k+1 \right ) \lambda \,b}} \right ) \]