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Added Feb. 11, 2019.
Problem Chapter 3.6.1.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b y^n w_y = c \sin (\lambda x) + k \sin (\mu y) \]
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*Sin[lambda*x] + k*Sin[mu*y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to \frac {a b \lambda \mu c_1\left (\frac {a y-b x}{a}\right )+a k \lambda \sin \left (\frac {b \mu x}{a}\right ) \sin \left (\frac {\mu (a y-b x)}{a}\right )-a k \lambda \cos \left (\frac {b \mu x}{a}\right ) \cos \left (\frac {\mu (a y-b x)}{a}\right )-b c \mu \cos (\lambda x)}{a b \lambda \mu }\right \}\right \} \]
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*diff(w(x,y),x) + b*diff(w(x,y),y) = c*sin(lambda*x)+k*sin(mu*y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[ w \left ( x,y \right ) =-{\frac {1}{b\mu \,a\lambda } \left ( -{\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) b\mu \,a\lambda +\cos \left ( \lambda \,x \right ) cb\mu +ka\cos \left ( {\frac { \left ( ya-bx \right ) \mu }{a}}+{\frac {b\mu \,x}{a}} \right ) \lambda \right ) } \]
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Added Feb. 11, 2019.
Problem Chapter 3.6.1.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b y^n w_y = c \sin (\lambda x+\mu y) \]
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 = a*D[w[x, y], x] + b*D[w[x, y], y] == c*Sin[lambda*x + mu*y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to \frac {-c \cos \left (\mu \left (\frac {a y-b x}{a}+\frac {b x}{a}\right )+\lambda x\right )+a \lambda c_1\left (\frac {a y-b x}{a}\right )+b \mu c_1\left (\frac {a y-b x}{a}\right )}{a \lambda +b \mu }\right \}\right \} \]
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*diff(w(x,y),x) + b*diff(w(x,y),y) = c*sin(lambda*x+mu*y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[ w \left ( x,y \right ) =-{\frac {c}{a\lambda +b\mu }\cos \left ( {\frac { \left ( a\lambda +b\mu \right ) x}{a}}+{\frac { \left ( ya-bx \right ) \mu }{a}} \right ) }+{\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) \]
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Added Feb. 11, 2019.
Problem Chapter 3.6.1.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ x w_x + y w_y = a x \sin (\lambda x+\mu y) \]
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*Sin[lambda*x + mu*y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to \frac {-a x \cos \left (x \left (\lambda +\frac {\mu y}{x}\right )\right )+\lambda x c_1\left (\frac {y}{x}\right )+\mu y c_1\left (\frac {y}{x}\right )}{\lambda x+\mu y}\right \}\right \} \]
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*sin(lambda*x+mu*y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[ w \left ( x,y \right ) =-{a\cos \left ( x \left ( {\frac {\mu \,y}{x}}+\lambda \right ) \right ) \left ( {\frac {\mu \,y}{x}}+\lambda \right ) ^{-1}}+{\it \_F1} \left ( {\frac {y}{x}} \right ) \]
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Added Feb. 11, 2019.
Problem Chapter 3.6.1.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b \sin ^n(\lambda x) w_y = c\sin ^m(\mu x)+s \sin ^k(\beta y) \]
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*Sin[lambda*x]^n*D[w[x, y], y] == c*Sin[mu*x]^m + s*Sin[beta*y]^k; 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*diff(w(x,y),x) + b*sin(lambda*x)^n*diff(w(x,y),y) = c*sin(mu*x)^m+s*sin(beta*y)^k; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[ w \left ( x,y \right ) =\int ^{x}\!{\frac {1}{a} \left ( s \left ( \sin \left ( {\frac {\beta }{a} \left ( \int \! \left ( \sin \left ( {\it \_b}\,\lambda \right ) \right ) ^{n}\,{\rm d}{\it \_b}b+ \left ( -\int \!{\frac {b \left ( \sin \left ( \lambda \,x \right ) \right ) ^{n}}{a}}\,{\rm d}x+y \right ) a \right ) } \right ) \right ) ^{k}+c \left ( \sin \left ( {\it \_b}\,\mu \right ) \right ) ^{m} \right ) }{d{\it \_b}}+{\it \_F1} \left ( -\int \!{\frac {b \left ( \sin \left ( \lambda \,x \right ) \right ) ^{n}}{a}}\,{\rm d}x+y \right ) \]
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Added Feb. 11, 2019.
Problem Chapter 3.6.1.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b \sin ^n(\lambda y) w_y = c\sin ^m(\mu x)+s \sin ^k(\beta y) \]
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*Sin[lambda*y]^n*D[w[x, y], y] == c*Sin[mu*x]^m + s*Sin[beta*y]^k; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to \frac {\sin ^2(\mu x)^{-\frac {m}{2}-\frac {1}{2}} \left (-c \cos (\mu x) \sin ^{m+1}(\mu x) \text {Hypergeometric2F1}\left (\frac {1}{2},\frac {1-m}{2},\frac {3}{2},\cos ^2(\mu x)\right )+a c_1() \mu \sin ^2(\mu x)^{\frac {m}{2}+\frac {1}{2}}+\mu s x \sin ^k(\beta y) \sin ^2(\mu x)^{\frac {m}{2}+\frac {1}{2}}\right )}{a \mu }\right \}\right \} \]
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*diff(w(x,y),x) + b*sin(lambda*y)^n*diff(w(x,y),y) = c*sin(mu*x)^m+s*sin(beta*y)^k; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[ w \left ( x,y \right ) =\int ^{y}\!{\frac { \left ( \sin \left ( {\it \_b}\,\lambda \right ) \right ) ^{-n}}{b} \left ( \left ( \sin \left ( {\frac {\mu \, \left ( a\int \! \left ( \sin \left ( {\it \_b}\,\lambda \right ) \right ) ^{-n}\,{\rm d}{\it \_b}+bx-a\int \! \left ( \sin \left ( y\lambda \right ) \right ) ^{-n}\,{\rm d}y \right ) }{b}} \right ) \right ) ^{m}c+s \left ( \sin \left ( \beta \,{\it \_b} \right ) \right ) ^{k} \right ) }{d{\it \_b}}+{\it \_F1} \left ( {\frac {bx-a\int \! \left ( \sin \left ( y\lambda \right ) \right ) ^{-n}\,{\rm d}y}{b}} \right ) \] Result has unresolved integrals