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Added Feb. 11, 2019.
Problem Chapter 3.5.2.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b w_y = c x^n+ s \ln ^k(\lambda y) \]
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 = a*D[w[x, y], x] + b*D[w[x, y], y] == c*x^n + s*Log[lambda*y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to \frac {a b n c_1\left (\frac {a y-b x}{a}\right )+a b c_1\left (\frac {a y-b x}{a}\right )+b n s x \log \left (\lambda \left (\frac {a y-b x}{a}+\frac {b x}{a}\right )\right )+b s x \log \left (\lambda \left (\frac {a y-b x}{a}+\frac {b x}{a}\right )\right )-b n s x \log (a y)-b s x \log (a y)+a n s y \log (a y)+a s y \log (a y)+b c x^{n+1}-b n s x-b s x}{a b (n+1)}\right \}\right \} \]
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*diff(w(x,y),x) + b*diff(w(x,y),y) = c*x^n+s*ln(lambda*y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[ w \left ( x,y \right ) ={\frac {y}{ab \left ( n+1 \right ) } \left ( \ln \left ( {\frac {bx\lambda }{a}}+{\frac { \left ( ya-bx \right ) \lambda }{a}} \right ) ans+\ln \left ( {\frac {bx\lambda }{a}}+{\frac { \left ( ya-bx \right ) \lambda }{a}} \right ) as-ans-as \right ) }+{\frac {1}{ab \left ( n+1 \right ) } \left ( {\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) abn+{x}^{n+1}cb+{\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) ba \right ) } \]
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Added Feb. 11, 2019.
Problem Chapter 3.5.2.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + a w_y = b y^2+c x^n y+ s \ln ^k(\lambda x) \]
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*D[w[x, y], y] == b*y^2 + c*x^n*y + s*Log[lambda*x]^k; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to \frac {(-\log (\lambda x))^{-k} \left (3 n^2 s \log ^k(\lambda x) \text {Gamma}(k+1,-\log (\lambda x))+9 n s \log ^k(\lambda x) \text {Gamma}(k+1,-\log (\lambda x))+6 s \log ^k(\lambda x) \text {Gamma}(k+1,-\log (\lambda x))+a^2 b \lambda n^2 x^3 (-\log (\lambda x))^k+3 a^2 b \lambda n x^3 (-\log (\lambda x))^k+2 a^2 b \lambda x^3 (-\log (\lambda x))^k-3 a b \lambda n^2 x^2 y (-\log (\lambda x))^k-9 a b \lambda n x^2 y (-\log (\lambda x))^k-6 a b \lambda x^2 y (-\log (\lambda x))^k+3 \lambda n^2 c_1(y-a x) (-\log (\lambda x))^k-3 a c \lambda x^{n+2} (-\log (\lambda x))^k+9 \lambda n c_1(y-a x) (-\log (\lambda x))^k+6 \lambda c_1(y-a x) (-\log (\lambda x))^k+3 b \lambda n^2 x y^2 (-\log (\lambda x))^k+9 b \lambda n x y^2 (-\log (\lambda x))^k+6 b \lambda x y^2 (-\log (\lambda x))^k+6 c \lambda y x^{n+1} (-\log (\lambda x))^k+3 c \lambda n y x^{n+1} (-\log (\lambda x))^k\right )}{3 \lambda (n+1) (n+2)}\right \}\right \} \]
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 := diff(w(x,y),x) + a*diff(w(x,y),y) = b*y^2+c*x^n*y+s*ln(lambda*x)^k; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[ w \left ( x,y \right ) =\int ^{x}\!b{{\it \_a}}^{2}{a}^{2}+ca{{\it \_a}}^{n+1}+2\, \left ( -ax+y \right ) ab{\it \_a}+{{\it \_a}}^{n} \left ( -ax+y \right ) c+ \left ( -ax+y \right ) ^{2}b+s \left ( \ln \left ( \lambda \,{\it \_a} \right ) \right ) ^{k}{d{\it \_a}}+{\it \_F1} \left ( -ax+y \right ) \] Result has unresolved integrals
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Added Feb. 11, 2019.
Problem Chapter 3.5.2.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + a w_y = b ln^k(\lambda x) \ln ^n(\beta y) \]
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 = D[w[x, y], x] + a*D[w[x, y], y] == b*Log[lambda*x]^k*Log[beta*y]^n; 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 := diff(w(x,y),x) + a*diff(w(x,y),y) = b*ln(lambda*x)^k*ln(beta*y)^n; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[ w \left ( x,y \right ) =\int ^{x}\!b \left ( \ln \left ( \lambda \,{\it \_a} \right ) \right ) ^{k} \left ( \ln \left ( \beta \, \left ( {\it \_a}\,a-ax+y \right ) \right ) \right ) ^{n}{d{\it \_a}}+{\it \_F1} \left ( -ax+y \right ) \]