51 HFOPDE, chapter 2.5.2

____________________________________________________________________________________

51.1 problem number 1

problem number 425

Added January 14, 2019.

Problem 2.5.2.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ w_x + a x^n \ln ^k(\lambda y) w_y = 0 \]

Mathematica ✓

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d]; 
 pde = D[w[x, y], x] + a*x^n*Log[lambda*y]^k*D[w[x, y], y] == 0; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[ \left \{\left \{w(x,y)\to c_1\left (\frac {\frac {n (-\log (\lambda y))^k \log ^{-k}(\lambda y) \text {Gamma}(1-k,-\log (\lambda y))}{\lambda }+\frac {(-\log (\lambda y))^k \log ^{-k}(\lambda y) \text {Gamma}(1-k,-\log (\lambda y))}{\lambda }-a x^{n+1}}{n+1}\right )\right \}\right \} \]

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'; C:='C';lambda:='lambda';s:='s';B:='B';mu:='mu';d:='d'; 
pde := diff(w(x,y),x)+a*x^n*ln(lambda*y)^k*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {{x}^{n+1}a-n\int \! \left ( \ln \left ( y\lambda \right ) \right ) ^{-k}\,{\rm d}y-\int \! \left ( \ln \left ( y\lambda \right ) \right ) ^{-k}\,{\rm d}y}{a}} \right ) \]

____________________________________________________________________________________

51.2 problem number 2

problem number 426

Added January 14, 2019.

Problem 2.5.2.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ w_x + a y^n \ln ^k(\lambda x) w_y = 0 \]

Mathematica ✓

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d]; 
 pde = D[w[x, y], x] + a*y^n*Log[lambda*x]^k*D[w[x, y], y] == 0; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[ \left \{\left \{w(x,y)\to c_1\left (-\frac {y^{-n} (-\log (\lambda x))^{-k} \left (-a y^n \log ^k(\lambda x) \text {Gamma}(k+1,-\log (\lambda x))+a n y^n \log ^k(\lambda x) \text {Gamma}(k+1,-\log (\lambda x))+\lambda y (-\log (\lambda x))^k\right )}{\lambda (n-1)}\right )\right \}\right \} \]

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'; C:='C';lambda:='lambda';s:='s';B:='B';mu:='mu';d:='d'; 
pde := diff(w(x,y),x)+a*y^n*ln(lambda*x)^k*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {y}{{y}^{n}}}+an\int \! \left ( \ln \left ( \lambda \,x \right ) \right ) ^{k}\,{\rm d}x-a\int \! \left ( \ln \left ( \lambda \,x \right ) \right ) ^{k}\,{\rm d}x \right ) \]

____________________________________________________________________________________

51.3 problem number 3

problem number 427

Added January 14, 2019.

Problem 2.5.2.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ w_x + \left (y^2+ a \ln (\beta x) y - a b \ln (\beta x) - b^2 \right ) w_y = 0 \]

Mathematica ✗

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d]; 
 pde = D[w[x, y], x] + (y^2 + a*Log[beta*x]*y - a*b*Log[beta*x] - b^2)*D[w[x, y], y] == 0; 
 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'; C:='C';lambda:='lambda';s:='s';B:='B';mu:='mu';d:='d'; 
pde := diff(w(x,y),x)+(y^2+ a*ln(beta*x)* y - a*b*ln(beta*x) - b^2)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac { \left ( \beta \,x \right ) ^{ax}{{\rm e}^{- \left ( a-2\,b \right ) x}}+y\int \! \left ( \beta \,x \right ) ^{ax}{{\rm e}^{- \left ( a-2\,b \right ) x}}\,{\rm d}x-b\int \! \left ( \beta \,x \right ) ^{ax}{{\rm e}^{- \left ( a-2\,b \right ) x}}\,{\rm d}x}{-b+y}} \right ) \]

____________________________________________________________________________________

51.4 problem number 4

problem number 428

Added January 14, 2019.

Problem 2.5.2.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ w_x + \left (y^2+ a x \ln ^m(b x) y + a \ln ^m(b x) \right ) w_y = 0 \]

Mathematica ✗

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d]; 
 pde = D[w[x, y], x] + (y^2 + a*x*Log[b*x]^m*y + a*Log[b*x]^m)*D[w[x, y], y] == 0; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[ \text {Failed} \]

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'; C:='C';lambda:='lambda';s:='s';B:='B';mu:='mu';d:='d'; 
pde := diff(w(x,y),x)+(y^2+ a*x*ln(b*x)^m * y + a *ln(b*x)^m)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1}{yx+1} \left ( yx\int \!{{\rm e}^{\int \!{\frac {a \left ( \ln \left ( bx \right ) \right ) ^{m}{x}^{2}-2}{x}}\,{\rm d}x}}\,{\rm d}x+{{\rm e}^{\int \!{\frac {a \left ( \ln \left ( bx \right ) \right ) ^{m}{x}^{2}-2}{x}}\,{\rm d}x}}x+\int \!{{\rm e}^{\int \!{\frac {a \left ( \ln \left ( bx \right ) \right ) ^{m}{x}^{2}-2}{x}}\,{\rm d}x}}\,{\rm d}x \right ) } \right ) \]

____________________________________________________________________________________

51.5 problem number 5

problem number 429

Added January 14, 2019.

Problem 2.5.2.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ w_x + \left (a x^n y^2- a b x^{n+1} y \ln (x) + b \ln (x) + b \right ) w_y = 0 \]

Mathematica ✗

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d]; 
 pde = D[w[x, y], x] + (a*x^n*y^2 - a*b*x^(n + 1)*y*Log[x] + b*Log[x] + b)*D[w[x, y], y] == 0; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[ \text {Failed} \]

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'; C:='C';lambda:='lambda';s:='s';B:='B';mu:='mu';d:='d'; 
pde := diff(w(x,y),x)+(a*x^n*y^2- a*b*x^(n+1)*y*ln(x)  + b*ln(x) + b)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[ \text { sol=() } \]

____________________________________________________________________________________

51.6 problem number 6

problem number 430

Added January 14, 2019.

Problem 2.5.2.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ w_x - \left ((n+1)x^n y^2 - a x^{n+1}(\ln x)^m y + a(\ln x)^m \right ) w_y = 0 \]

Mathematica ✗

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d]; 
 pde = D[w[x, y], x] - ((n + 1)*x^n*y^2 - a*x^(n + 1)*Log[x]^m*y + a*Log[x]^m)*D[w[x, y], y] == 0; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[ \text {Failed} \]

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'; C:='C';lambda:='lambda';s:='s';B:='B';mu:='mu';d:='d'; 
pde := diff(w(x,y),x)-((n+1)*x^n*y^2 - a*x^(n+1)*ln(x)^m*y + a*ln(x)^m)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1}{y{x}^{n+1}-1} \left ( y{x}^{n+1}\int \!{\frac {{{\rm e}^{a\int \!{x}^{n+1} \left ( \ln \left ( x \right ) \right ) ^{m}\,{\rm d}x-2\,n\ln \left ( x \right ) }}{x}^{n}}{{x}^{2}}}\,{\rm d}xn+y{x}^{n+1}\int \!{\frac {{{\rm e}^{a\int \!{x}^{n+1} \left ( \ln \left ( x \right ) \right ) ^{m}\,{\rm d}x-2\,n\ln \left ( x \right ) }}{x}^{n}}{{x}^{2}}}\,{\rm d}x-{{\rm e}^{\int \!{\frac {a{x}^{n+1} \left ( \ln \left ( x \right ) \right ) ^{m}x-2\,n-2}{x}}\,{\rm d}x}}{x}^{n+1}-\int \!{\frac {{{\rm e}^{a\int \!{x}^{n+1} \left ( \ln \left ( x \right ) \right ) ^{m}\,{\rm d}x-2\,n\ln \left ( x \right ) }}{x}^{n}}{{x}^{2}}}\,{\rm d}xn-\int \!{\frac {{{\rm e}^{a\int \!{x}^{n+1} \left ( \ln \left ( x \right ) \right ) ^{m}\,{\rm d}x-2\,n\ln \left ( x \right ) }}{x}^{n}}{{x}^{2}}}\,{\rm d}x \right ) } \right ) \]

____________________________________________________________________________________

51.7 problem number 7

problem number 431

Added January 14, 2019.

Problem 2.5.2.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ w_x + \left (a (\ln x)^n y^2 + b m x^{m-1} - a b^2 x^{2 m} (\ln x)^n \right ) w_y = 0 \]

Mathematica ✗

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d]; 
 pde = D[w[x, y], x] + (a*Log[x]^n*y^2 + b*m*x^(m - 1) - a*b^2*x^(2*m)*Log[x]^n)*D[w[x, y], y] == 0; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[ \text {Failed} \]

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'; C:='C';lambda:='lambda';s:='s';B:='B';mu:='mu';d:='d'; 
pde := diff(w(x,y),x)+(a *ln(x)^n*y^2 + b*m*x^(m-1) - a*b^2*x^(2*m)* ln(x)^n)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[ \text { sol=() } \]

____________________________________________________________________________________

51.8 problem number 8

problem number 432

Added January 14, 2019.

Problem 2.5.2.8 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ w_x + \left (a (\ln x)^n y^2 - a b x y(\ln x)^{n+1} + b \ln x+ b \right ) w_y = 0 \]

Mathematica ✗

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d]; 
 pde = D[w[x, y], x] + (a*Log[x]^n*y^2 - a*b*x*y*Log[x]^(n + 1) + b*Log[x] + b)*D[w[x, y], y] == 0; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[ \text {Failed} \]

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'; C:='C';lambda:='lambda';s:='s';B:='B';mu:='mu';d:='d'; 
pde := diff(w(x,y),x)+(a*ln(x)^n*y^2 - a*b*x*y*(ln(x))^(n+1) + b*ln(x)+ b )*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[ \text { sol=() } \]

____________________________________________________________________________________

51.9 problem number 9

problem number 433

Added January 14, 2019.

Problem 2.5.2.9 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ w_x + \left (a (\ln x)^k (y - b x^n-c)^3 + b n x^{n-1} \right ) w_y = 0 \]

Mathematica ✓

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d]; 
 pde = D[w[x, y], x] + (a*Log[x]^k*(y - b*x^n - c)^3 + b*n*x^(n - 1))*D[w[x, y], y] == 0; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[ \left \{\left \{w(x,y)\to c_1\left (\frac {(-\log (x))^{-k} \left (2 a b^2 x^{2 n} \log ^k(x) \text {Gamma}(k+1,-\log (x))+4 a b c x^n \log ^k(x) \text {Gamma}(k+1,-\log (x))-4 a b y x^n \log ^k(x) \text {Gamma}(k+1,-\log (x))+2 a c^2 \log ^k(x) \text {Gamma}(k+1,-\log (x))-4 a c y \log ^k(x) \text {Gamma}(k+1,-\log (x))+2 a y^2 \log ^k(x) \text {Gamma}(k+1,-\log (x))+(-\log (x))^k\right )}{\left (b x^n+c-y\right )^2}\right )\right \}\right \} \]

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'; C:='C';lambda:='lambda';s:='s';B:='B';mu:='mu';d:='d'; 
pde := diff(w(x,y),x)+(a*(ln(x))^k*(y - b*x^n-c)^3 + b*n*x^(n-1) ) *diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {2\,{x}^{2\,n}a{b}^{2}\int \! \left ( \ln \left ( x \right ) \right ) ^{k}\,{\rm d}x-4\,{x}^{n}yab\int \! \left ( \ln \left ( x \right ) \right ) ^{k}\,{\rm d}x+4\,{x}^{n}abc\int \! \left ( \ln \left ( x \right ) \right ) ^{k}\,{\rm d}x+2\,{y}^{2}a\int \! \left ( \ln \left ( x \right ) \right ) ^{k}\,{\rm d}x-4\,yac\int \! \left ( \ln \left ( x \right ) \right ) ^{k}\,{\rm d}x+2\,a{c}^{2}\int \! \left ( \ln \left ( x \right ) \right ) ^{k}\,{\rm d}x+1}{ \left ( b{x}^{n}+c-y \right ) ^{2}}} \right ) \]

____________________________________________________________________________________

51.10 problem number 10

problem number 434

Added January 14, 2019.

Problem 2.5.2.10 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ w_x + \left (a (\ln x)^n y^2 + b(\ln x)^m y+ b c (\ln x)^m - a c^2 (\ln x)^n \right ) w_y = 0 \]

Mathematica ✗

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d]; 
 pde = D[w[x, y], x] + (a*Log[x]^n*y^2 + b*Log[x]^m*y + b*c*Log[x]^m - a*c^2*Log[x]^n)*D[w[x, y], y] == 0; 
 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'; C:='C';lambda:='lambda';s:='s';B:='B';mu:='mu';d:='d'; 
pde := diff(w(x,y),x)+(a*(ln(x))^n*y^2 + b*(ln(x))^m *y+ b*c* (ln(x))^m - a*c^2* (ln(x))^n) *diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac {ya\int \! \left ( \ln \left ( x \right ) \right ) ^{n}{{\rm e}^{-2\,ca\int \! \left ( \ln \left ( x \right ) \right ) ^{n}\,{\rm d}x+b\int \! \left ( \ln \left ( x \right ) \right ) ^{m}\,{\rm d}x}}\,{\rm d}x+a\int \! \left ( \ln \left ( x \right ) \right ) ^{n}{{\rm e}^{-2\,ca\int \! \left ( \ln \left ( x \right ) \right ) ^{n}\,{\rm d}x+b\int \! \left ( \ln \left ( x \right ) \right ) ^{m}\,{\rm d}x}}\,{\rm d}xc+{{\rm e}^{-2\,ca\int \! \left ( \ln \left ( x \right ) \right ) ^{n}\,{\rm d}x+b\int \! \left ( \ln \left ( x \right ) \right ) ^{m}\,{\rm d}x}}}{c+y}} \right ) \]

____________________________________________________________________________________

51.11 problem number 11

problem number 435

Added January 14, 2019.

Problem 2.5.2.11 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ x w_x + \left (a y+ b \ln x \right )^2 w_y = 0 \]

Mathematica ✓

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d]; 
 pde = x*D[w[x, y], x] + (a*y + b*Log[x])^2*D[w[x, y], y] == 0; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[ \left \{\left \{w(x,y)\to c_1\left (\tan ^{-1}\left (\frac {a^3 y \sqrt {\frac {b}{a^3}}+a^2 b \sqrt {\frac {b}{a^3}} \log (x)}{b}\right )-a^2 \sqrt {\frac {b}{a^3}} \log (x)\right )\right \}\right \} \]

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'; C:='C';lambda:='lambda';s:='s';B:='B';mu:='mu';d:='d'; 
pde := x*diff(w(x,y),x)+(a*y+ b*ln(x))^2 *diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1}{a\sqrt {ab}} \left ( -\ln \left ( x \right ) \sqrt {ab}+\arctan \left ( {\frac {a \left ( ya+b\ln \left ( x \right ) \right ) }{\sqrt {ab}}} \right ) \right ) } \right ) \]

____________________________________________________________________________________

51.12 problem number 12

problem number 436

Added January 14, 2019.

Problem 2.5.2.12 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ x w_x + \left (x y^2 - A^2 x (\ln \beta x)^2 + A \right ) w_y = 0 \]

Mathematica ✗

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d]; 
 pde = x*D[w[x, y], x] + (x*y^2 - A^2*x*Log[beta*x]^2 + A)*D[w[x, y], y] == 0; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[ \text {Failed} \]

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'; C:='C';lambda:='lambda';s:='s';B:='B';mu:='mu';d:='d'; 
pde := x*diff(w(x,y),x)+(x*y^2 - A^2*x*(ln(beta*x))^2 + A) *diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[ \text { sol=() } \]

____________________________________________________________________________________

51.13 problem number 13

problem number 437

Added January 14, 2019.

Problem 2.5.2.13 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ x w_x + \left (x y^2 - A^2 x (\ln (\beta x))^{2 k} + k A (\ln (\beta x))^{k-1} \right ) w_y = 0 \]

Mathematica ✗

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d]; 
 pde = x*D[w[x, y], x] + (x*y^2 - A^2*x*Log[beta*x]^(2*k) + k*A*Log[beta*x]^(k - 1))*D[w[x, y], y] == 0; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[ \text {Failed} \]

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'; C:='C';lambda:='lambda';s:='s';B:='B';mu:='mu';d:='d'; 
pde := x*diff(w(x,y),x)+(x*y^2 - A^2*x*(ln(beta*x))^(2*k) + k*A*(ln(beta*x))^(k-1))*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[ \text { sol=() } \]

____________________________________________________________________________________

51.14 problem number 14

problem number 438

Added January 14, 2019.

Problem 2.5.2.14 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ x w_x + \left (a x^n y^2 + b - a b^2 x^n (\ln x)^2 \right ) w_y = 0 \]

Mathematica ✗

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d]; 
 pde = x*D[w[x, y], x] + (a*x^n*y^2 + b - a*b^2*x^n*Log[x]^2)*D[w[x, y], y] == 0; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[ \text {Failed} \]

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'; C:='C';lambda:='lambda';s:='s';B:='B';mu:='mu';d:='d'; 
pde := x*diff(w(x,y),x)+(a*x^n*y^2 + b - a*b^2*x^n*(ln(x))^2)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[ \text { sol=() } \]

____________________________________________________________________________________

51.15 problem number 15

problem number 439

Added January 14, 2019.

Problem 2.5.2.15 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ x w_x + \left ( a (\ln (\lambda x))^m y^2 + k y+ a b^2 x^{2 k} (\ln (\lambda x))^m \right ) w_y = 0 \]

Mathematica ✓

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d]; 
 pde = x*D[w[x, y], x] + (a*Log[lambda*x]^m*y^2 + k*y + a*b^2*x^(2*k)*Log[lambda*x]^m)*D[w[x, y], y] == 0; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[ \left \{\left \{w(x,y)\to c_1\left (\tan ^{-1}\left (\frac {y x^{-k}}{\sqrt {b^2}}\right )-\frac {a \sqrt {b^2} x^k (\lambda x)^{-k} \log ^m(\lambda x) (-k \log (\lambda x))^{-m} \text {Gamma}(m+1,-k \log (\lambda x))}{k}\right )\right \}\right \} \]

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'; C:='C';lambda:='lambda';s:='s';B:='B';mu:='mu';d:='d'; 
pde := x*diff(w(x,y),x)+(a*(ln(lambda*x))^m*y^2 + k*y+ a*b^2*x^(2*k)* (ln(lambda*x))^m )*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( ab\int \! \left ( \ln \left ( \lambda \,x \right ) \right ) ^{m}{x}^{k-1}\,{\rm d}x-\arctan \left ( {\frac {{x}^{-k}y}{b}} \right ) \right ) \]

____________________________________________________________________________________

51.16 problem number 16

problem number 440

Added January 14, 2019.

Problem 2.5.2.16 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ x w_x + \left ( a x^n(y + b \ln x)^2 - b \right ) w_y = 0 \]

Mathematica ✓

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d]; 
 pde = x*D[w[x, y], x] + (a*x^n*(y + b*Log[x])^2 - b)*D[w[x, y], y] == 0; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[ \left \{\left \{w(x,y)\to c_1\left (\frac {a b x^n \log (x)+a y x^n+n}{n (b \log (x)+y)}\right )\right \}\right \} \]

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'; C:='C';lambda:='lambda';s:='s';B:='B';mu:='mu';d:='d'; 
pde := x*diff(w(x,y),x)+(a*x^n*(y + b*ln(x))^2 - b)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {ab\ln \left ( x \right ) {x}^{n}+a{x}^{n}y+n}{n \left ( y+b\ln \left ( x \right ) \right ) }} \right ) \]

____________________________________________________________________________________

51.17 problem number 17

problem number 441

Added January 14, 2019.

Problem 2.5.2.17 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ x w_x + \left ( a x^{2 n} \ln (x) y^2 + (b x^n \ln x - n) y + c \ln x \right ) w_y = 0 \]

Mathematica ✓

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d]; 
 pde = x*D[w[x, y], x] + (a*x^(2*n)*Log[x]*y^2 + (b*x^n*Log[x] - n)*y + c*Log[x])*D[w[x, y], y] == 0; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[ \left \{\left \{w(x,y)\to c_1\left (\frac {\left (\sqrt {b^2-4 a c}+2 a y x^n+b\right ) \exp \left (\frac {\sqrt {a} \sqrt {c} x^n \left (\frac {\sqrt {b^2-4 a c}}{\sqrt {a} \sqrt {c}}+\frac {b}{\sqrt {a} \sqrt {c}}\right ) (n \log (x)-1)}{2 n^2}-\frac {\sqrt {a} \sqrt {c} x^n \left (\frac {b}{\sqrt {a} \sqrt {c}}-\frac {\sqrt {b^2-4 a c}}{\sqrt {a} \sqrt {c}}\right ) (n \log (x)-1)}{2 n^2}\right )}{\sqrt {b^2-4 a c}-2 a y x^n-b}\right )\right \}\right \} \]

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'; C:='C';lambda:='lambda';s:='s';B:='B';mu:='mu';d:='d'; 
pde := x*diff(w(x,y),x)+(a*x^(2*n)*ln(x)* y^2 + (b* x^n *ln(x) - n)*y + c *ln(x))*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {b}{\sqrt {{b}^{2} \left ( 4\,ca-{b}^{2} \right ) }{n}^{2}} \left ( -\ln \left ( x \right ) {x}^{n}\sqrt {{b}^{2} \left ( 4\,ca-{b}^{2} \right ) }n+2\,b{n}^{2}\arctan \left ( {\frac {b \left ( 2\,a{x}^{n}y+b \right ) }{\sqrt {4\,ac{b}^{2}-{b}^{4}}}} \right ) +\sqrt {{b}^{2} \left ( 4\,ca-{b}^{2} \right ) }{x}^{n} \right ) } \right ) \]

____________________________________________________________________________________

51.18 problem number 18

problem number 442

Added January 14, 2019.

Problem 2.5.2.18 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ x^k w_x + \left (a y^n (\ln x)^m + b y (\ln x)^s \right ) w_y = 0 \]

Mathematica ✗

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d]; 
 pde = x^k*D[w[x, y], x] + (a*y^n*Log[x]^m + b*y*Log[x]^s)*D[w[x, y], y] == 0; 
 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'; C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s'; 
pde := x^k*diff(w(x,y),x)+(a*y^n*(ln(x))^m + b*y*(ln(x))^s )*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( {y}^{-n+1}{{\rm e}^{b\int \!{x}^{-k} \left ( \ln \left ( x \right ) \right ) ^{s}\,{\rm d}x \left ( n-1 \right ) }}+an\int \!{{\rm e}^{b\int \!{x}^{-k} \left ( \ln \left ( x \right ) \right ) ^{s}\,{\rm d}x \left ( n-1 \right ) }}{x}^{-k} \left ( \ln \left ( x \right ) \right ) ^{m}\,{\rm d}x-a\int \!{{\rm e}^{b\int \!{x}^{-k} \left ( \ln \left ( x \right ) \right ) ^{s}\,{\rm d}x \left ( n-1 \right ) }}{x}^{-k} \left ( \ln \left ( x \right ) \right ) ^{m}\,{\rm d}x \right ) \]

____________________________________________________________________________________

51.19 problem number 19

problem number 443

Added January 14, 2019.

Problem 2.5.2.19 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ (a \ln x+b) w_x + \left (y^2+ c(\ln x)^n y- \lambda ^2 + \lambda c( \ln x)^n \right ) w_y = 0 \]

Mathematica ✗

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d]; 
 pde = (a*Log[x] + b)*D[w[x, y], x] + (y^2 + c*Log[x]^n*y - lambda^2 + lambda*c*Log[x]^n)*D[w[x, y], y] == 0; 
 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'; C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s'; 
pde := (a*ln(x)+b)*diff(w(x,y),x)+(y^2+ c*(ln(x))^n*y- lambda^2 + lambda*c*(ln(x))^n )*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{1 \left ( \int \!{\frac {1}{\ln \left ( x \right ) a+b}{{\rm e}^{\int \!{\frac {c \left ( \ln \left ( x \right ) \right ) ^{n}-2\,\lambda }{\ln \left ( x \right ) a+b}}\,{\rm d}x}}}\,{\rm d}x\lambda \,{{\rm e}^{\int \!{\frac {c \left ( \ln \left ( x \right ) \right ) ^{n}-2\,\lambda }{\ln \left ( x \right ) a+b}}\,{\rm d}x+\int \!-{\frac {c \left ( \ln \left ( x \right ) \right ) ^{n}-2\,\lambda }{\ln \left ( x \right ) a+b}}\,{\rm d}x}}+y\int \!{\frac {1}{\ln \left ( x \right ) a+b}{{\rm e}^{\int \!{\frac {c \left ( \ln \left ( x \right ) \right ) ^{n}-2\,\lambda }{\ln \left ( x \right ) a+b}}\,{\rm d}x}}}\,{\rm d}x+{{\rm e}^{\int \!{\frac {c \left ( \ln \left ( x \right ) \right ) ^{n}-2\,\lambda }{\ln \left ( x \right ) a+b}}\,{\rm d}x}} \right ) \left ( {{\rm e}^{\int \!{\frac {c \left ( \ln \left ( x \right ) \right ) ^{n}-2\,\lambda }{\ln \left ( x \right ) a+b}}\,{\rm d}x+\int \!-{\frac {c \left ( \ln \left ( x \right ) \right ) ^{n}-2\,\lambda }{\ln \left ( x \right ) a+b}}\,{\rm d}x}}\lambda +y \right ) ^{-1}} \right ) \]

____________________________________________________________________________________

51.20 problem number 20

problem number 444

Added January 14, 2019.

Problem 2.5.2.20 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ (a \ln x+b) w_x + \left ((\ln x)^n y^2- c y - \lambda ^2 ( \ln x)^n + c \lambda \right ) w_y = 0 \]

Mathematica ✗

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d]; 
 pde = (a*Log[x] + b)*D[w[x, y], x] + (Log[x]^n*y^2 - c*y - lambda^2*Log[x]^n + c*lambda)*D[w[x, y], y] == 0; 
 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'; C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s'; 
pde := (a*ln(x)+b)*diff(w(x,y),x)+((ln(x))^n*y^2- c*y - lambda^2*(ln(x))^n + c*lambda )*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{1 \left ( \int \!{\frac { \left ( \ln \left ( x \right ) \right ) ^{n}}{\ln \left ( x \right ) a+b}{{\rm e}^{\int \!{\frac {2\, \left ( \ln \left ( x \right ) \right ) ^{n}\lambda -c}{\ln \left ( x \right ) a+b}}\,{\rm d}x}}}\,{\rm d}x\lambda \,{{\rm e}^{\int \!{\frac {2\, \left ( \ln \left ( x \right ) \right ) ^{n}\lambda -c}{\ln \left ( x \right ) a+b}}\,{\rm d}x+\int \!-{\frac {2\, \left ( \ln \left ( x \right ) \right ) ^{n}\lambda -c}{\ln \left ( x \right ) a+b}}\,{\rm d}x}}-y\int \!{\frac { \left ( \ln \left ( x \right ) \right ) ^{n}}{\ln \left ( x \right ) a+b}{{\rm e}^{\int \!{\frac {2\, \left ( \ln \left ( x \right ) \right ) ^{n}\lambda -c}{\ln \left ( x \right ) a+b}}\,{\rm d}x}}}\,{\rm d}x-{{\rm e}^{\int \!{\frac {2\, \left ( \ln \left ( x \right ) \right ) ^{n}\lambda -c}{\ln \left ( x \right ) a+b}}\,{\rm d}x}} \right ) \left ( {{\rm e}^{\int \!{\frac {2\, \left ( \ln \left ( x \right ) \right ) ^{n}\lambda -c}{\ln \left ( x \right ) a+b}}\,{\rm d}x+\int \!-{\frac {2\, \left ( \ln \left ( x \right ) \right ) ^{n}\lambda -c}{\ln \left ( x \right ) a+b}}\,{\rm d}x}}\lambda -y \right ) ^{-1}} \right ) \]

____________________________________________________________________________________

51.21 problem number 21

problem number 445

Added January 14, 2019.

Problem 2.5.2.21 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ x^2 \ln (a x) w_x - \left ( x^2 y^2 \ln (a x) + 1\right ) w_y = 0 \]

Mathematica ✗

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d]; 
 pde = x^2*Log[a*x]*D[w[x, y], x] - (x^2*y^2*Log[a*x] + 1)*D[w[x, y], y] == 0; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[ \text {Failed} \]

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'; C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s'; 
pde := x^2*ln(a*x)*diff(w(x,y),x)-(x^2*y^2* ln(a*x)+ 1  )*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {xy\ln \left ( ax \right ) -1}{\ln \left ( ax \right ) \Ei \left ( 1,-\ln \left ( ax \right ) \right ) yx+a{x}^{2}y-\Ei \left ( 1,-\ln \left ( ax \right ) \right ) }} \right ) \]

____________________________________________________________________________________

51.22 problem number 22

problem number 446

Added January 14, 2019.

Problem 2.5.2.22 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ \ln ^k(\lambda x) w_x + \left ( a y^n + b y \ln ^m x\right ) w_y = 0 \]

Mathematica ✗

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d]; 
 pde = Log[lambda*x]^k*D[w[x, y], x] + (a*y^n + b*y*Log[x]^m)*D[w[x, y], y] == 0; 
 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'; C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s'; 
pde :=  (ln(lambda*x))^k*diff(w(x,y),x)+(a*y^n+ b*y* (ln(x))^m )*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( {y}^{-n+1}{{\rm e}^{b\int \! \left ( \ln \left ( x \right ) \right ) ^{m} \left ( \ln \left ( \lambda \,x \right ) \right ) ^{-k}\,{\rm d}x \left ( n-1 \right ) }}+an\int \!{{\rm e}^{b\int \! \left ( \ln \left ( x \right ) \right ) ^{m} \left ( \ln \left ( \lambda \,x \right ) \right ) ^{-k}\,{\rm d}x \left ( n-1 \right ) }} \left ( \ln \left ( \lambda \,x \right ) \right ) ^{-k}\,{\rm d}x-a\int \!{{\rm e}^{b\int \! \left ( \ln \left ( x \right ) \right ) ^{m} \left ( \ln \left ( \lambda \,x \right ) \right ) ^{-k}\,{\rm d}x \left ( n-1 \right ) }} \left ( \ln \left ( \lambda \,x \right ) \right ) ^{-k}\,{\rm d}x \right ) \]

____________________________________________________________________________________

51.23 problem number 23

problem number 447

Added January 14, 2019.

Problem 2.5.2.23 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ \ln ^k(\lambda x) w_x + \left ( a y^n \ln ^m x + b y \right ) w_y = 0 \]

Mathematica ✗

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d]; 
 pde = Log[lambda*x]^k*D[w[x, y], x] + (a*y^n*Log[x]^m + b*y)*D[w[x, y], y] == 0; 
 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'; C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s'; 
pde :=  (ln(lambda*x))^k*diff(w(x,y),x)+(a*y^n*(ln(x))^m+ b*y )*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( {y}^{-n+1}{{\rm e}^{b\int \! \left ( \ln \left ( \lambda \,x \right ) \right ) ^{-k}\,{\rm d}x \left ( n-1 \right ) }}+an\int \!{{\rm e}^{b\int \! \left ( \ln \left ( \lambda \,x \right ) \right ) ^{-k}\,{\rm d}x \left ( n-1 \right ) }} \left ( \ln \left ( \lambda \,x \right ) \right ) ^{-k} \left ( \ln \left ( x \right ) \right ) ^{m}\,{\rm d}x-a\int \!{{\rm e}^{b\int \! \left ( \ln \left ( \lambda \,x \right ) \right ) ^{-k}\,{\rm d}x \left ( n-1 \right ) }} \left ( \ln \left ( \lambda \,x \right ) \right ) ^{-k} \left ( \ln \left ( x \right ) \right ) ^{m}\,{\rm d}x \right ) \]