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]];
\[\left \{\left \{w(x,y)\to c_1\left (b a^{-n-1} \text {Gamma}(n+1,a x)+y e^{-a x}\right )+\frac {c \log ^k(\lambda x) (-\log (\lambda x))^{-k} \text {Gamma}(k+1,-\log (\lambda x))}{\lambda }\right \}\right \}\]
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'));
\[w \left ( x,y \right ) =\int \!c \left ( \ln \left ( \lambda \,x \right ) \right ) ^{k}\,{\rm d}x+{\it \_F1} \left ( {\frac { \left ( -{x}^{n} \left ( ax \right ) ^{-n/2} \WhittakerM \left ( n/2,n/2+1/2,ax \right ) {{\rm e}^{1/2\,ax}}b+ay \left ( n+1 \right ) \right ) {{\rm e}^{-ax}}}{a \left ( n+1 \right ) }} \right ) \] 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["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]];
\[\left \{\left \{w(x,y)\to c_1\left (y x^{-\frac {b}{a}}\right )+\frac {x^k (a k m \log (y)+a k n \log (x)-a n-b m)}{a^2 k^2}\right \}\right \}\]
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'));
\[w \left ( x,y \right ) =1/2\,{\frac {1}{{k}^{2}{a}^{2}} \left ( \left ( 2\,\ln \left ( x \right ) akn+ \left ( -m \left ( \left ( i \left ( {\it csgn} \left ( iy \right ) \right ) ^{3}-i \left ( {\it csgn} \left ( iy \right ) \right ) ^{2}{\it csgn} \left ( iy{x}^{-{\frac {b}{a}}} \right ) -i \left ( {\it csgn} \left ( iy \right ) \right ) ^{2}{\it csgn} \left ( i{x}^{{\frac {b}{a}}} \right ) +i{\it csgn} \left ( iy \right ) {\it csgn} \left ( iy{x}^{-{\frac {b}{a}}} \right ) {\it csgn} \left ( i{x}^{{\frac {b}{a}}} \right ) \right ) \pi -2\,\ln \left ( {x}^{{\frac {b}{a}}} \right ) -2\,\ln \left ( y{x}^{-{\frac {b}{a}}} \right ) \right ) k-2\,n \right ) a-2\,bm \right ) {x}^{k}+2\,{\it \_F1} \left ( y{x}^{-{\frac {b}{a}}} \right ) {k}^{2}{a}^{2} \right ) }\]
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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["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'));
\[w \left ( x,y \right ) =\int ^{x}\!{\frac {{{\it \_a}}^{-k}}{a} \left ( c\ln \left ( {\it \_a}\,\lambda \right ) +s \left ( \ln \left ( \beta \, \left ( {\frac {-{x}^{1-k}b \left ( n-1 \right ) +{y}^{-n+1}a \left ( k-1 \right ) +{{\it \_a}}^{1-k}b \left ( n-1 \right ) }{a \left ( k-1 \right ) }} \right ) ^{- \left ( n-1 \right ) ^{-1}} \right ) \right ) ^{l} \right ) }{d{\it \_a}}+{\it \_F1} \left ( {\frac {-{x}^{1-k}b \left ( n-1 \right ) +{y}^{-n+1}a \left ( k-1 \right ) }{a \left ( k-1 \right ) }} \right ) \]
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