Added April 8, 2019.
Problem Chapter 5.6.1.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ a w_x + b w_y = c w + k \sin (\lambda x+\mu y) \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*D[w[x, y], y] == c*w[x,y]+ k*Sin[lambda*x+mu*y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to -\frac {k ((a \lambda +b \mu ) \cos (\lambda x+\mu y)+c \sin (\lambda x+\mu y))}{(a \lambda +b \mu )^2+c^2}+e^{\frac {c x}{a}} c_1\left (y-\frac {b x}{a}\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x)+ b*diff(w(x,y),y) = c*w(x,y)+k*sin(lambda*x+mu*y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \frac {\left (-\left (c \sin \left (\lambda x +\mu y \right )+\left (a \lambda +\mu b \right ) \cos \left (\lambda x +\mu y \right )\right ) k \,{\mathrm e}^{-\frac {c x}{a}}+\left (\lambda ^{2} a^{2}+2 a b \lambda \mu +b^{2} \mu ^{2}+c^{2}\right ) \textit {\_F1} \left (\frac {a y -b x}{a}\right )\right ) {\mathrm e}^{\frac {c x}{a}}}{\lambda ^{2} a^{2}+2 a b \lambda \mu +b^{2} \mu ^{2}+c^{2}}\]
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Added April 8, 2019.
Problem Chapter 5.6.1.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ a w_x + b w_y = w + c_1 \sin ^k(\lambda x)+c_2 \sin ^n(\beta y) \]
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*Sin[beta*y]^n; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to e^{\frac {x}{a}} c_1\left (y-\frac {b x}{a}\right )-i \left (\frac {\text {c1} \left (-1+e^{2 i \lambda x}\right ) \sin ^k(\lambda x) \, _2F_1\left (1,\frac {1}{2} \left (k+\frac {i}{a \lambda }+2\right );\frac {1}{2} \left (-k+\frac {i}{a \lambda }+2\right );e^{2 i \lambda x}\right )}{a k \lambda -i}+\frac {\text {c2} \left (-1+e^{2 i \beta y}\right ) \sin ^n(\beta y) \, _2F_1\left (1,\frac {1}{2} \left (n+\frac {i}{b \beta }+2\right );\frac {1}{2} \left (-n+\frac {i}{b \beta }+2\right );e^{2 i \beta y}\right )}{b \beta n-i}\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x)+ b*diff(w(x,y),y) = w(x,y)+c1*sin(lambda*x)^k+c2*sin(beta*y)^n; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \left (\int _{}^{x}\frac {\left (\mathit {c1} \left (\sin ^{k}\left (\textit {\_a} \lambda \right )\right )+\mathit {c2} \left (\sin ^{n}\left (\frac {\left (a y -\left (-\textit {\_a} +x \right ) b \right ) \beta }{a}\right )\right )\right ) {\mathrm e}^{-\frac {\textit {\_a}}{a}}}{a}d \textit {\_a} +\textit {\_F1} \left (\frac {a y -b x}{a}\right )\right ) {\mathrm e}^{\frac {x}{a}}\]
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Added April 8, 2019.
Problem Chapter 5.6.1.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ a w_x + b w_y = c w +\sin ^k(\lambda x) \sin ^n(\beta y) \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*D[w[x, y], y] == c*w[x,y]+ Sin[lambda*x]^k*Sin[beta*y]^n; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to e^{\frac {c x}{a}} \left (\int _1^x\frac {e^{-\frac {c K[1]}{a}} \sin ^k(\lambda K[1]) \sin ^n\left (\beta \left (y+\frac {b (K[1]-x)}{a}\right )\right )}{a}dK[1]+c_1\left (y-\frac {b x}{a}\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x)+ b*diff(w(x,y),y) = c*w(x,y)+sin(lambda*x)^k*sin(beta*y)^n; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \left (\int _{}^{x}\frac {\left (\sin ^{k}\left (\textit {\_a} \lambda \right )\right ) \left (\sin ^{n}\left (\frac {\left (a y -\left (-\textit {\_a} +x \right ) b \right ) \beta }{a}\right )\right ) {\mathrm e}^{-\frac {\textit {\_a} c}{a}}}{a}d \textit {\_a} +\textit {\_F1} \left (\frac {a y -b x}{a}\right )\right ) {\mathrm e}^{\frac {c x}{a}}\]
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Added April 8, 2019.
Problem Chapter 5.6.1.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ a x w_x + b y w_y = c w +k \sin (\lambda x+\mu y) \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*x*D[w[x, y], x] + b*y*D[w[x, y], y] == c*w[x,y]+ k*Sin[lambda*x+beta*y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to x^{\frac {c}{a}} \left (\int _1^x\frac {k K[1]^{-\frac {a+c}{a}} \sin \left (\beta y K[1]^{\frac {b}{a}} x^{-\frac {b}{a}}+\lambda K[1]\right )}{a}dK[1]+c_1\left (y x^{-\frac {b}{a}}\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := a*x*diff(w(x,y),x)+ b*y*diff(w(x,y),y) = c*w(x,y)+k*sin(lambda*x+beta*y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \left (\int _{}^{x}\frac {k \,\textit {\_a}^{\frac {-a -c}{a}} \sin \left (\beta y \,\textit {\_a}^{\frac {b}{a}} x^{-\frac {b}{a}}+\textit {\_a} \lambda \right )}{a}d \textit {\_a} +\textit {\_F1} \left (y \,x^{-\frac {b}{a}}\right )\right ) x^{\frac {c}{a}}\]
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Added April 8, 2019.
Problem Chapter 5.6.1.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ x w_x + y w_y = a x \sin (\lambda x+\mu y) w + b \sin (\nu x) \]
Mathematica ✓
ClearAll["Global`*"]; pde = x*D[w[x, y], x] + y*D[w[x, y], y] == a*x*Sin[lambda*x+beta*y]*w[x,y]+ b*Sin[nu*x]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to e^{-\frac {a x \cos (\beta y+\lambda x)}{\beta y+\lambda x}} \left (\int _1^x\frac {b e^{\frac {a x \cos \left (\left (\lambda +\frac {\beta y}{x}\right ) K[1]\right )}{\lambda x+\beta y}} \sin (\nu K[1])}{K[1]}dK[1]+c_1\left (\frac {y}{x}\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := x*diff(w(x,y),x)+ b*diff(w(x,y),y) = a*x*sin(lambda*x+beta*y)*w(x,y)+ b*sin(nu*x); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \left (\int _{}^{x}\frac {b \,{\mathrm e}^{-a \left (\int \sin \left (b \beta \ln \left (\textit {\_b} \right )+\textit {\_b} \lambda +\left (-b \ln \left (x \right )+y \right ) \beta \right )d \textit {\_b} \right )} \sin \left (\textit {\_b} \nu \right )}{\textit {\_b}}d \textit {\_b} +\textit {\_F1} \left (-b \ln \left (x \right )+y \right )\right ) {\mathrm e}^{\int _{}^{x}a \sin \left (b \beta \ln \left (\textit {\_a} \right )+\textit {\_a} \lambda +\left (-b \ln \left (x \right )+y \right ) \beta \right )d \textit {\_a}}\]
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Added April 8, 2019.
Problem Chapter 5.6.1.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ a \sin ^n(\lambda x) w_x + b \sin ^m(\mu x) w_y = c \sin ^k(\nu x) w + p \sin ^s(\beta y) \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*Sin[lambda*x]^n*D[w[x, y], x] + b*Sin[mu*x]^m*D[w[x, y], y] == c*Sin[nu*x]^k*w[x,y]+ p*Sin[beta*y]^s; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \exp \left (\int _1^x\frac {c \sin ^{-n}(\lambda K[2]) \sin ^k(\nu K[2])}{a}dK[2]\right ) \left (\int _1^x\frac {\exp \left (-\int _1^{K[3]}\frac {c \sin ^{-n}(\lambda K[2]) \sin ^k(\nu K[2])}{a}dK[2]\right ) p \sin ^{-n}(\lambda K[3]) \sin ^s\left (\beta \left (y-\int _1^x\frac {b \sin ^{-n}(\lambda K[1]) \sin ^m(\mu K[1])}{a}dK[1]+\int _1^{K[3]}\frac {b \sin ^{-n}(\lambda K[1]) \sin ^m(\mu K[1])}{a}dK[1]\right )\right )}{a}dK[3]+c_1\left (y-\int _1^x\frac {b \sin ^{-n}(\lambda K[1]) \sin ^m(\mu K[1])}{a}dK[1]\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := a*sin(lambda*x)^n*diff(w(x,y),x)+ b*sin(mu*x)^m*diff(w(x,y),y) = c*sin(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'));
\[w \left (x , y\right ) = \left (\int _{}^{x}\frac {p \left (\sin ^{-n}\left (\textit {\_f} \lambda \right )\right ) \left (\sin ^{s}\left (\frac {\left (a y +b \left (\int \left (\sin ^{m}\left (\textit {\_f} \mu \right )\right ) \left (\sin ^{-n}\left (\textit {\_f} \lambda \right )\right )d \textit {\_f} \right )-b \left (\int \left (\sin ^{-n}\left (\lambda x \right )\right ) \left (\sin ^{m}\left (\mu x \right )\right )d x \right )\right ) \beta }{a}\right )\right ) {\mathrm e}^{-\frac {c \left (\int \left (\sin ^{-n}\left (\textit {\_f} \lambda \right )\right ) \left (\sin ^{k}\left (\textit {\_f} \nu \right )\right )d \textit {\_f} \right )}{a}}}{a}d \textit {\_f} +\textit {\_F1} \left (\frac {a y -b \left (\int \left (\sin ^{-n}\left (\lambda x \right )\right ) \left (\sin ^{m}\left (\mu x \right )\right )d x \right )}{a}\right )\right ) {\mathrm e}^{\int \frac {c \left (\sin ^{-n}\left (\lambda x \right )\right ) \left (\sin ^{k}\left (\nu x \right )\right )}{a}d x}\]
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Added April 8, 2019.
Problem Chapter 5.6.1.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ a \sin ^n(\lambda x) w_x + b \sin ^m(\mu x) w_y = c \sin ^k(\nu y) w + p \sin ^s(\beta x) \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*Sin[lambda*x]^n*D[w[x, y], x] + b*Sin[mu*x]^m*D[w[x, y], y] == c*Sin[nu*y]^k*w[x,y]+ p*Sin[beta*x]^s; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \exp \left (\int _1^x\frac {c \sin ^{-n}(\lambda K[2]) \sin ^k\left (\nu \left (y-\int _1^x\frac {b \sin ^{-n}(\lambda K[1]) \sin ^m(\mu K[1])}{a}dK[1]+\int _1^{K[2]}\frac {b \sin ^{-n}(\lambda K[1]) \sin ^m(\mu K[1])}{a}dK[1]\right )\right )}{a}dK[2]\right ) \left (\int _1^x\frac {\exp \left (-\int _1^{K[3]}\frac {c \sin ^{-n}(\lambda K[2]) \sin ^k\left (\nu \left (y-\int _1^x\frac {b \sin ^{-n}(\lambda K[1]) \sin ^m(\mu K[1])}{a}dK[1]+\int _1^{K[2]}\frac {b \sin ^{-n}(\lambda K[1]) \sin ^m(\mu K[1])}{a}dK[1]\right )\right )}{a}dK[2]\right ) p \sin ^s(\beta K[3]) \sin ^{-n}(\lambda K[3])}{a}dK[3]+c_1\left (y-\int _1^x\frac {b \sin ^{-n}(\lambda K[1]) \sin ^m(\mu K[1])}{a}dK[1]\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := a*sin(lambda*x)^n*diff(w(x,y),x)+ b*sin(mu*x)^m*diff(w(x,y),y) = c*sin(nu*y)^k*w(x,y)+ p*sin(beta*x)^s; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \left (\int _{}^{x}\frac {p \left (\sin ^{s}\left (\textit {\_f} \beta \right )\right ) \left (\sin ^{-n}\left (\textit {\_f} \lambda \right )\right ) {\mathrm e}^{-\frac {c \left (\int \left (\sin ^{-n}\left (\textit {\_f} \lambda \right )\right ) \left (\sin ^{k}\left (\frac {\left (a y +b \left (\int \left (\sin ^{m}\left (\textit {\_f} \mu \right )\right ) \left (\sin ^{-n}\left (\textit {\_f} \lambda \right )\right )d \textit {\_f} \right )-b \left (\int \left (\sin ^{-n}\left (\lambda x \right )\right ) \left (\sin ^{m}\left (\mu x \right )\right )d x \right )\right ) \nu }{a}\right )\right )d \textit {\_f} \right )}{a}}}{a}d \textit {\_f} +\textit {\_F1} \left (\frac {a y -b \left (\int \left (\sin ^{-n}\left (\lambda x \right )\right ) \left (\sin ^{m}\left (\mu x \right )\right )d x \right )}{a}\right )\right ) {\mathrm e}^{\int _{}^{x}\frac {c \left (\sin ^{-n}\left (\textit {\_b} \lambda \right )\right ) \left (\sin ^{k}\left (\frac {\left (a y +a \left (\int \frac {b \left (\sin ^{-n}\left (\textit {\_b} \lambda \right )\right ) \left (\sin ^{m}\left (\textit {\_b} \mu \right )\right )}{a}d \textit {\_b} \right )-b \left (\int \left (\sin ^{-n}\left (\lambda x \right )\right ) \left (\sin ^{m}\left (\mu x \right )\right )d x \right )\right ) \nu }{a}\right )\right )}{a}d \textit {\_b}}\]
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