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'));
\[w \left ( x,y \right ) ={\frac {1}{{a}^{2}{\lambda }^{2}+2\,ab\lambda \,\mu +{b}^{2}{\mu }^{2}+{c}^{2}}{{\rm e}^{{\frac {cx}{a}}}} \left ( \left ( {a}^{2}{\lambda }^{2}+2\,ab\lambda \,\mu +{b}^{2}{\mu }^{2}+{c}^{2} \right ) {\it \_F1} \left ( {\frac {ay-xb}{a}} \right ) - \left ( \left ( a\lambda +\mu \,b \right ) \cos \left ( x\lambda +\mu \,y \right ) +c\sin \left ( x\lambda +\mu \,y \right ) \right ) k{{\rm e}^{-{\frac {cx}{a}}}} \right ) }\]
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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'));
\[w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac {1}{a} \left ( {\it c1}\, \left ( \sin \left ( {\it \_a}\,\lambda \right ) \right ) ^{k}+{\it c2}\, \left ( \sin \left ( {\frac {\beta \, \left ( ay-b \left ( x-{\it \_a} \right ) \right ) }{a}} \right ) \right ) ^{n} \right ) {{\rm e}^{-{\frac {{\it \_a}}{a}}}}}{d{\it \_a}}+{\it \_F1} \left ( {\frac {ay-xb}{a}} \right ) \right ) {{\rm e}^{{\frac {x}{a}}}}\]
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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'));
\[w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac { \left ( \sin \left ( {\it \_a}\,\lambda \right ) \right ) ^{k}}{a} \left ( \sin \left ( {\frac {\beta \, \left ( ay-b \left ( x-{\it \_a} \right ) \right ) }{a}} \right ) \right ) ^{n}{{\rm e}^{-{\frac {c{\it \_a}}{a}}}}}{d{\it \_a}}+{\it \_F1} \left ( {\frac {ay-xb}{a}} \right ) \right ) {{\rm e}^{{\frac {cx}{a}}}}\]
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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'));
\[w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac {k}{a}{{\it \_a}}^{{\frac {-a-c}{a}}}\sin \left ( \beta \,y{x}^{-{\frac {b}{a}}}{{\it \_a}}^{{\frac {b}{a}}}+{\it \_a}\,\lambda \right ) }{d{\it \_a}}+{\it \_F1} \left ( y{x}^{-{\frac {b}{a}}} \right ) \right ) {x}^{{\frac {c}{a}}}\]
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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'));
\[w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac {\sin \left ( \nu \,{\it \_b} \right ) b{{\rm e}^{-a\int \!\sin \left ( \ln \left ( {\it \_b} \right ) b\beta +\beta \, \left ( -b\ln \left ( x \right ) +y \right ) +{\it \_b}\,\lambda \right ) \,{\rm d}{\it \_b}}}}{{\it \_b}}}{d{\it \_b}}+{\it \_F1} \left ( -b\ln \left ( x \right ) +y \right ) \right ) {{\rm e}^{\int ^{x}\!\sin \left ( \ln \left ( {\it \_a} \right ) b\beta +\beta \, \left ( -b\ln \left ( x \right ) +y \right ) +{\it \_a}\,\lambda \right ) a{d{\it \_a}}}}\]
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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'));
\[w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac {p \left ( \sin \left ( {\it \_f}\,\lambda \right ) \right ) ^{-n}}{a} \left ( \sin \left ( {\frac {\beta \, \left ( b\int \! \left ( \sin \left ( \mu \,{\it \_f} \right ) \right ) ^{m} \left ( \sin \left ( {\it \_f}\,\lambda \right ) \right ) ^{-n}\,{\rm d}{\it \_f}-b\int \! \left ( \sin \left ( \mu \,x \right ) \right ) ^{m} \left ( \sin \left ( x\lambda \right ) \right ) ^{-n}\,{\rm d}x+ay \right ) }{a}} \right ) \right ) ^{s}{{\rm e}^{-{\frac {c\int \! \left ( \sin \left ( \nu \,{\it \_f} \right ) \right ) ^{k} \left ( \sin \left ( {\it \_f}\,\lambda \right ) \right ) ^{-n}\,{\rm d}{\it \_f}}{a}}}}}{d{\it \_f}}+{\it \_F1} \left ( {\frac {-b\int \! \left ( \sin \left ( \mu \,x \right ) \right ) ^{m} \left ( \sin \left ( x\lambda \right ) \right ) ^{-n}\,{\rm d}x+ay}{a}} \right ) \right ) {{\rm e}^{\int \!{\frac { \left ( \sin \left ( \nu \,x \right ) \right ) ^{k}c \left ( \sin \left ( x\lambda \right ) \right ) ^{-n}}{a}}\,{\rm d}x}}\]
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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'));
\[w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac {p \left ( \sin \left ( \beta \,{\it \_f} \right ) \right ) ^{s} \left ( \sin \left ( {\it \_f}\,\lambda \right ) \right ) ^{-n}}{a}{{\rm e}^{-{\frac {c}{a}\int \! \left ( \sin \left ( {\frac {\nu \, \left ( b\int \! \left ( \sin \left ( \mu \,{\it \_f} \right ) \right ) ^{m} \left ( \sin \left ( {\it \_f}\,\lambda \right ) \right ) ^{-n}\,{\rm d}{\it \_f}-b\int \! \left ( \sin \left ( \mu \,x \right ) \right ) ^{m} \left ( \sin \left ( x\lambda \right ) \right ) ^{-n}\,{\rm d}x+ay \right ) }{a}} \right ) \right ) ^{k} \left ( \sin \left ( {\it \_f}\,\lambda \right ) \right ) ^{-n}\,{\rm d}{\it \_f}}}}}{d{\it \_f}}+{\it \_F1} \left ( {\frac {-b\int \! \left ( \sin \left ( \mu \,x \right ) \right ) ^{m} \left ( \sin \left ( x\lambda \right ) \right ) ^{-n}\,{\rm d}x+ay}{a}} \right ) \right ) {{\rm e}^{\int ^{x}\!{\frac {c \left ( \sin \left ( {\it \_b}\,\lambda \right ) \right ) ^{-n}}{a} \left ( \sin \left ( {\frac {\nu }{a} \left ( \int \!{\frac {b \left ( \sin \left ( \mu \,{\it \_b} \right ) \right ) ^{m} \left ( \sin \left ( {\it \_b}\,\lambda \right ) \right ) ^{-n}}{a}}\,{\rm d}{\it \_b}a+ay-b\int \! \left ( \sin \left ( \mu \,x \right ) \right ) ^{m} \left ( \sin \left ( x\lambda \right ) \right ) ^{-n}\,{\rm d}x \right ) } \right ) \right ) ^{k}}{d{\it \_b}}}}\]
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