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["Global`*"]; 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]];
\[\left \{\left \{w(x,y)\to \frac {c \log ^k(\lambda x) (-\log (\lambda x))^{-k} \operatorname {Gamma}(k+1,-\log (\lambda x))}{\lambda }+c_1\left (b a^{-n-1} \operatorname {Gamma}(n+1,a x)+y e^{-a x}\right )\right \}\right \}\]
Maple ✓
restart; 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'));
\[w \left (x , y\right ) = \int c \ln \left (\lambda x \right )^{k}d x +\mathit {\_F1} \left (\frac {\left (-b x^{n} \left (a x \right )^{-\frac {n}{2}} \WhittakerM \left (\frac {n}{2}, \frac {n}{2}+\frac {1}{2}, a x \right ) {\mathrm e}^{\frac {a x}{2}}+\left (n +1\right ) a y \right ) {\mathrm e}^{-a x}}{\left (n +1\right ) a}\right )\] 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["Global`*"]; 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]];
\[\left \{\left \{w(x,y)\to \frac {x^k (a k m \log (y)+a k n \log (x)-a n-b m)}{a^2 k^2}+c_1\left (y x^{-\frac {b}{a}}\right )\right \}\right \}\]
Maple ✓
restart; 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'));
\[w \left (x , y\right ) = \frac {2 a^{2} k^{2} \mathit {\_F1} \left (y x^{-\frac {b}{a}}\right )+\left (2 a k n \ln \left (x \right )-2 b m +\left (-\left (-2 \ln \left (x^{\frac {b}{a}}\right )-2 \ln \left (y x^{-\frac {b}{a}}\right )+\left (i \mathrm {csgn}\left (i y \right )^{3}-i \mathrm {csgn}\left (i y \right )^{2} \mathrm {csgn}\left (i x^{\frac {b}{a}}\right )-i \mathrm {csgn}\left (i y \right )^{2} \mathrm {csgn}\left (i y x^{-\frac {b}{a}}\right )+i \mathrm {csgn}\left (i y \right ) \mathrm {csgn}\left (i x^{\frac {b}{a}}\right ) \mathrm {csgn}\left (i y x^{-\frac {b}{a}}\right )\right ) \pi \right ) k m -2 n \right ) a \right ) x^{k}}{2 a^{2} k^{2}}\]
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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["Global`*"]; 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]];
\[\left \{\left \{w(x,y)\to \int _1^x\frac {K[1]^{-k} \left (s \log ^l\left (\beta \left (\frac {a (k-1) x^k y^n K[1]^k}{a (k-1) x^k y K[1]^k-b (n-1) y^n \left (x K[1]^k-x^k K[1]\right )}\right )^{\frac {1}{n-1}}\right )+c \log ^m(\lambda K[1])\right )}{a}dK[1]+c_1\left (\frac {b x^{1-k}}{a (k-1)}-\frac {y^{1-n}}{n-1}\right )\right \}\right \}\]
Maple ✓
restart; 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'));
\[w \left (x , y\right ) = \int _{}^{x}\frac {\left (c \ln \left (\mathit {\_a} \lambda \right )+s \ln \left (\beta \left (\frac {\left (k -1\right ) a y^{-n +1}+\left (n -1\right ) b \mathit {\_a}^{-k +1}-\left (n -1\right ) b x^{-k +1}}{\left (k -1\right ) a}\right )^{-\frac {1}{n -1}}\right )^{l}\right ) \mathit {\_a}^{-k}}{a}d\mathit {\_a} +\mathit {\_F1} \left (\frac {\left (k -1\right ) a y^{-n +1}-\left (n -1\right ) b x^{-k +1}}{\left (k -1\right ) a}\right )\]
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