Added April 11, 2019.
Problem Chapter 5.6.5.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b w_y = w + c_1 \sin ^k(\lambda x)+c_2 \cos ^n(\beta y) \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*D[w[x, y], y] == w[x,y]+c1*Sin[lambda*x]^k+c2*Cos[beta*y]^n; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to e^{\frac {x}{a}} c_1\left (y-\frac {b x}{a}\right )-\frac {i \text {c1} \left (-1+e^{2 i \lambda x}\right ) \sin ^k(\lambda x) \, _2F_1\left (1,\frac {1}{2} \left (k+\frac {i}{a \lambda }+2\right );\frac {1}{2} \left (-k+\frac {i}{a \lambda }+2\right );e^{2 i \lambda x}\right )}{a k \lambda -i}+\frac {\text {c2} \left (1+e^{2 i \beta y}\right ) \cos ^n(\beta y) \, _2F_1\left (1,\frac {1}{2} \left (n+\frac {i}{b \beta }+2\right );\frac {1}{2} \left (-n+\frac {i}{b \beta }+2\right );-e^{2 i \beta y}\right )}{-1-i b \beta n}\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x)+ b*diff(w(x,y),y) = w(x,y)+c1*sin(lambda*x)^k+c2*cos(beta*y)^n; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \left (\int _{}^{x}\frac {\left (\mathit {c1} \left (\sin ^{k}\left (\textit {\_a} \lambda \right )\right )+\mathit {c2} \left (\cos ^{n}\left (\frac {\left (a y -\left (-\textit {\_a} +x \right ) b \right ) \beta }{a}\right )\right )\right ) {\mathrm e}^{-\frac {\textit {\_a}}{a}}}{a}d \textit {\_a} +\textit {\_F1} \left (\frac {a y -b x}{a}\right )\right ) {\mathrm e}^{\frac {x}{a}}\]
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Added April 11, 2019.
Problem Chapter 5.6.5.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b w_y = c w + \sin ^k(\lambda x) \cos ^n(\beta y) \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*D[w[x, y], y] == c*w[x,y]+Sin[lambda*x]^k*Cos[beta*y]^n; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to e^{\frac {c x}{a}} \left (\int _1^x\frac {e^{-\frac {c K[1]}{a}} \cos ^n\left (\beta \left (y+\frac {b (K[1]-x)}{a}\right )\right ) \sin ^k(\lambda K[1])}{a}dK[1]+c_1\left (y-\frac {b x}{a}\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x)+ b*diff(w(x,y),y) = c*w(x,y)+sin(lambda*x)^k*cos(beta*y)^n; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \left (\int _{}^{x}\frac {\left (\cos ^{n}\left (\frac {\left (a y -\left (-\textit {\_a} +x \right ) b \right ) \beta }{a}\right )\right ) \left (\sin ^{k}\left (\textit {\_a} \lambda \right )\right ) {\mathrm e}^{-\frac {\textit {\_a} c}{a}}}{a}d \textit {\_a} +\textit {\_F1} \left (\frac {a y -b x}{a}\right )\right ) {\mathrm e}^{\frac {c x}{a}}\]
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Added April 11, 2019.
Problem Chapter 5.6.5.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b \sin (\mu y) w_y = c \sin (\lambda x) w + k \cos (\nu x) + s \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*Sin[mu*y]*D[w[x, y], y] == c*Sin[lambda*x]*w[x,y]+k*Cos[nu*x]+s; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to e^{-\frac {c \cos (\lambda x)}{a \lambda }} \left (\int _1^x\frac {e^{\frac {c \cos (\lambda K[1])}{a \lambda }} (s+k \cos (\nu K[1]))}{a}dK[1]+c_1\left (\frac {\log \left (\tan \left (\frac {\mu y}{2}\right )\right )}{\mu }-\frac {b x}{a}\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x)+ b*sin(mu*y)*diff(w(x,y),y) = c*sin(lambda*x)*w(x,y)+k*cos(nu*x)+s; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \left (\int \frac {\left (k \cos \left (\nu x \right )+s \right ) {\mathrm e}^{\frac {c \cos \left (\lambda x \right )}{a \lambda }}}{a}d x +\textit {\_F1} \left (\frac {a \ln \left (\RootOf \left (\mu y -\arctan \left (\frac {2 \textit {\_Z} \,{\mathrm e}^{\frac {b \mu x}{a}}}{\textit {\_Z}^{2} {\mathrm e}^{\frac {2 b \mu x}{a}}+1}, -\frac {\textit {\_Z}^{2} {\mathrm e}^{\frac {2 b \mu x}{a}}-1}{\textit {\_Z}^{2} {\mathrm e}^{\frac {2 b \mu x}{a}}+1}\right )\right )\right )}{b \mu }\right )\right ) {\mathrm e}^{-\frac {c \cos \left (\lambda x \right )}{a \lambda }}\]
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Added April 11, 2019.
Problem Chapter 5.6.5.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b \sin (\mu y) w_y = c \sin (\lambda x) w + k \tan (\nu x) + s \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*Sin[mu*y]*D[w[x, y], y] == c*Sin[lambda*x]*w[x,y]+k*Tan[nu*x]+s; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to e^{-\frac {c \cos (\lambda x)}{a \lambda }} \left (\int _1^x\frac {e^{\frac {c \cos (\lambda K[1])}{a \lambda }} (s+k \tan (\nu K[1]))}{a}dK[1]+c_1\left (\frac {\log \left (\tan \left (\frac {\mu y}{2}\right )\right )}{\mu }-\frac {b x}{a}\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x)+ b*sin(mu*y)*diff(w(x,y),y) = c*sin(lambda*x)*w(x,y)+k*tan(nu*x)+s; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \left (\int \frac {\left (k \sin \left (\nu x \right )+s \cos \left (\nu x \right )\right ) {\mathrm e}^{\frac {c \cos \left (\lambda x \right )}{a \lambda }}}{a \cos \left (\nu x \right )}d x +\textit {\_F1} \left (\frac {a \ln \left (\RootOf \left (\mu y -\arctan \left (\frac {2 \textit {\_Z} \,{\mathrm e}^{\frac {b \mu x}{a}}}{\textit {\_Z}^{2} {\mathrm e}^{\frac {2 b \mu x}{a}}+1}, -\frac {\textit {\_Z}^{2} {\mathrm e}^{\frac {2 b \mu x}{a}}-1}{\textit {\_Z}^{2} {\mathrm e}^{\frac {2 b \mu x}{a}}+1}\right )\right )\right )}{b \mu }\right )\right ) {\mathrm e}^{-\frac {c \cos \left (\lambda x \right )}{a \lambda }}\]
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Added April 11, 2019.
Problem Chapter 5.6.5.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b \tan (\mu y) w_y = c \tan (\lambda x) w + k \cot (\nu x) + s \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*Tan[mu*y]*D[w[x, y], y] == c*Tan[lambda*x]*w[x,y]+k*Cot[nu*x]+s; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \cos ^{-\frac {c}{a \lambda }}(\lambda x) \left (\int _1^x\frac {\cos ^{\frac {c}{a \lambda }}(\lambda K[1]) (s+k \cot (\nu K[1]))}{a}dK[1]+c_1\left (\frac {\log (\sin (\mu y))}{\mu }-\frac {b x}{a}\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x)+ b*tan(mu*y)*diff(w(x,y),y) = c*tan(lambda*x)*w(x,y)+k*cot(nu*x)+s; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \left (\int \frac {\left (k \cos \left (\nu x \right )+s \sin \left (\nu x \right )\right ) \left (\cos ^{\frac {c}{a \lambda }}\left (\lambda x \right )\right )}{a \sin \left (\nu x \right )}d x +\textit {\_F1} \left (\frac {-b \mu x +a \ln \left (\frac {\tan \left (\mu y \right )}{\sqrt {\tan ^{2}\left (\mu y \right )+1}}\right )}{b \mu }\right )\right ) \left (\cos ^{-\frac {c}{a \lambda }}\left (\lambda x \right )\right )\]
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Added April 11, 2019.
Problem Chapter 5.6.5.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a \sin ^n(\lambda x) w_x + b \cos ^m(\mu x) w_y = c \cos ^k(\nu x) w + p \sin ^s(\beta y) \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*Sin[lambda*x]^n*D[w[x, y], x] + b*Cos[mu*x]^m*D[w[x, y], y] == c*Cos[nu*x]^k*w[x,y]+p*Sin[beta*y]^s; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \exp \left (\int _1^x\frac {c \cos ^k(\nu K[2]) \sin ^{-n}(\lambda K[2])}{a}dK[2]\right ) \left (\int _1^x\frac {\exp \left (-\int _1^{K[3]}\frac {c \cos ^k(\nu K[2]) \sin ^{-n}(\lambda K[2])}{a}dK[2]\right ) p \sin ^{-n}(\lambda K[3]) \sin ^s\left (\beta \left (y-\int _1^x\frac {b \cos ^m(\mu K[1]) \sin ^{-n}(\lambda K[1])}{a}dK[1]+\int _1^{K[3]}\frac {b \cos ^m(\mu K[1]) \sin ^{-n}(\lambda K[1])}{a}dK[1]\right )\right )}{a}dK[3]+c_1\left (y-\int _1^x\frac {b \cos ^m(\mu K[1]) \sin ^{-n}(\lambda K[1])}{a}dK[1]\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := a*sin(lambda*x)^n*diff(w(x,y),x)+ b*cos(mu*x)^m*diff(w(x,y),y) = c*cos(nu*x)^k*w(x,y)+p*sin(beta*y)^s; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \left (\int _{}^{x}\frac {p \left (\sin ^{-n}\left (\textit {\_f} \lambda \right )\right ) \left (\sin ^{s}\left (\frac {\left (a y +b \left (\int \left (\cos ^{m}\left (\textit {\_f} \mu \right )\right ) \left (\sin ^{-n}\left (\textit {\_f} \lambda \right )\right )d \textit {\_f} \right )-b \left (\int \left (\cos ^{m}\left (\mu x \right )\right ) \left (\sin ^{-n}\left (\lambda x \right )\right )d x \right )\right ) \beta }{a}\right )\right ) {\mathrm e}^{-\frac {c \left (\int \left (\cos ^{k}\left (\textit {\_f} \nu \right )\right ) \left (\sin ^{-n}\left (\textit {\_f} \lambda \right )\right )d \textit {\_f} \right )}{a}}}{a}d \textit {\_f} +\textit {\_F1} \left (\frac {a y -b \left (\int \left (\cos ^{m}\left (\mu x \right )\right ) \left (\sin ^{-n}\left (\lambda x \right )\right )d x \right )}{a}\right )\right ) {\mathrm e}^{\int \frac {c \left (\cos ^{k}\left (\nu x \right )\right ) \left (\sin ^{-n}\left (\lambda x \right )\right )}{a}d x}\]
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Added April 11, 2019.
Problem Chapter 5.6.5.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a \tan ^n(\lambda x) w_x + b \cot ^m(\mu x) w_y = c \tan ^k(\nu x) w + p \cot ^s(\beta x) \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*Tan[lambda*x]^n*D[w[x, y], x] + b*Cot[mu*x]^m*D[w[x, y], y] == c*Tan[nu*x]^k*w[x,y]+p*Cot[beta*x]^s; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \exp \left (\int _1^x\frac {c \tan ^{-n}(\lambda K[2]) \tan ^k(\nu K[2])}{a}dK[2]\right ) \left (\int _1^x\frac {\exp \left (-\int _1^{K[3]}\frac {c \tan ^{-n}(\lambda K[2]) \tan ^k(\nu K[2])}{a}dK[2]\right ) p \cot ^s(\beta K[3]) \tan ^{-n}(\lambda K[3])}{a}dK[3]+c_1\left (y-\int _1^x\frac {b \cot ^m(\mu K[1]) \tan ^{-n}(\lambda K[1])}{a}dK[1]\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := a*tan(lambda*x)^n*diff(w(x,y),x)+ b*cot(mu*x)^m*diff(w(x,y),y) = c*tan(nu*x)^k*w(x,y)+p*cot(beta*x)^s; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \left (\int \frac {p \left (\frac {\cos \left (\beta x \right )}{\sin \left (\beta x \right )}\right )^{s} \left (\frac {\sin \left (\lambda x \right )}{\cos \left (\lambda x \right )}\right )^{-n} {\mathrm e}^{-\frac {c \left (\int \left (\frac {\sin \left (\lambda x \right )}{\cos \left (\lambda x \right )}\right )^{-n} \left (\frac {\sin \left (\nu x \right )}{\cos \left (\nu x \right )}\right )^{k}d x \right )}{a}}}{a}d x +\textit {\_F1} \left (\frac {a y -b \left (\int \left (\frac {\cos \left (\mu x \right )}{\sin \left (\mu x \right )}\right )^{m} \left (\frac {\sin \left (\lambda x \right )}{\cos \left (\lambda x \right )}\right )^{-n}d x \right )}{a}\right )\right ) {\mathrm e}^{\int \frac {c \left (\frac {\sin \left (\nu x \right )}{\cos \left (\nu x \right )}\right )^{k} \left (\tan ^{-n}\left (\lambda x \right )\right )}{a}d x}\]
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