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Added Feb. 11, 2019.
Problem Chapter 3.5.2.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + (a y+b x^n) w_y = c ln^k(\lambda x) \]
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*y + b*x^n)*D[w[x, y], y] == c*Log[lambda*x]^k; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to \frac {(-\log (\lambda x))^{-k} \left (\lambda (-\log (\lambda x))^k c_1\left (a^{-n-1} e^{-a x} \left (b e^{a x} \text {Gamma}(n+1,a x)+y a^{n+1}\right )\right )+c \log ^k(\lambda x) \text {Gamma}(k+1,-\log (\lambda x))\right )}{\lambda }\right \}\right \} \]
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*y+b*x^n)*diff(w(x,y),y) = c*ln(lambda*x)^k; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[ w \left ( x,y \right ) =\int \!c \left ( \ln \left ( \lambda \,x \right ) \right ) ^{k}\,{\rm d}x+{\it \_F1} \left ( -{\frac {{{\rm e}^{-ax}} \left ( \left ( ax \right ) ^{-n/2} \WhittakerM \left ( n/2,n/2+1/2,ax \right ) {x}^{n}{{\rm e}^{1/2\,ax}}b-any-ya \right ) }{a \left ( n+1 \right ) }} \right ) \] Result has unresolved integrals
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Added Feb. 11, 2019.
Problem Chapter 3.5.2.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a x w_x + b y w_y = x^k ( n \ln x+ m \ln y) \]
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*x*D[w[x, y], x] + b*y*D[w[x, y], y] == x^k*(n*Log[x] + m*Log[y]); sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to \frac {a^2 k^2 c_1\left (y x^{-\frac {b}{a}}\right )+a k m x^k \log (y)-a n x^k+a k n x^k \log (x)-b m x^k}{a^2 k^2}\right \}\right \} \]
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 := a*x*diff(w(x,y),x) + b*y*diff(w(x,y),y) = x^k*(n*ln(x)+m*ln(y)); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[ w \left ( x,y \right ) =1/2\,{\frac {1}{{k}^{2}{a}^{2}} \left ( i\pi \,{x}^{k}{\it csgn} \left ( i{x}^{{\frac {b}{a}}} \right ) \left ( {\it csgn} \left ( iy \right ) \right ) ^{2}akm-i\pi \,{x}^{k}{\it csgn} \left ( i{x}^{{\frac {b}{a}}} \right ) {\it csgn} \left ( iy \right ) {\it csgn} \left ( iy{x}^{-{\frac {b}{a}}} \right ) akm-i\pi \,{x}^{k} \left ( {\it csgn} \left ( iy \right ) \right ) ^{3}akm+i\pi \,{x}^{k} \left ( {\it csgn} \left ( iy \right ) \right ) ^{2}{\it csgn} \left ( iy{x}^{-{\frac {b}{a}}} \right ) akm+2\,\ln \left ( x \right ) {x}^{k}akn+2\,\ln \left ( {x}^{{\frac {b}{a}}} \right ) {x}^{k}mka+2\,{x}^{k}\ln \left ( y{x}^{-{\frac {b}{a}}} \right ) akm+2\,{\it \_F1} \left ( y{x}^{-{\frac {b}{a}}} \right ) {k}^{2}{a}^{2}-2\,{x}^{k}an-2\,{x}^{k}bm \right ) } \]
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Added Feb. 11, 2019.
Problem Chapter 3.5.2.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a x^k w_x + b y^n w_y = c \ln ^m(\lambda x)+s \ln ^l(\beta y) \]
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*x^k*D[w[x, y], x] + b*y^n*D[w[x, y], y] == c*Log[lambda*x]^m + s*Log[beta*y]^l; 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*x^k*diff(w(x,y),x) + b*y^n*diff(w(x,y),y) = c*ln(lambda*x)+s*ln(beta*y)^l; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime')); sol:=simplify(sol);
\[ w \left ( x,y \right ) =\int ^{x}\!{\frac {{{\it \_a}}^{-k}}{a} \left ( c\ln \left ( \lambda \,{\it \_a} \right ) +s \left ( \ln \left ( \beta \, \left ( {\frac {b \left ( n-1 \right ) {{\it \_a}}^{1-k}-{x}^{1-k}b \left ( n-1 \right ) +a{y}^{-n+1} \left ( k-1 \right ) }{a \left ( k-1 \right ) }} \right ) ^{- \left ( n-1 \right ) ^{-1}} \right ) \right ) ^{l} \right ) }{d{\it \_a}}+{\it \_F1} \left ( {\frac {-{x}^{1-k}b \left ( n-1 \right ) +a{y}^{-n+1} \left ( k-1 \right ) }{a \left ( k-1 \right ) }} \right ) \]