137 HFOPDE, chapter 5.5.2

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137.1 Problem 1

problem number 1091

Added April 8, 2019.

Problem Chapter 5.5.2.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b w_y = w + c_1 x^k+ c_2 \ln ^n(\beta y) \]

Mathematica ✓

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; 
 ClearAll[lambda, B, mu, d, g, B, v, f, h, q, p, delta, t]; 
 ClearAll[g1, g0, h2, h1, h0, f1, f2,sigma,lambda1,lambda2,n1,n2]; 
 ClearAll[a1, a0, b2, b1, b0, c2, c1, c0, k0, k1, k2, s1, s0, k22, k11, k12, s11, s22, s12, nu]; 
 pde = a*D[w[x, y], x] + b*D[w[x, y], y] == w[x,y]+c1*x^k+c2*Log[beta*y]^n; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; 
 sol = Simplify[sol];
 

\[ \left \{\left \{w(x,y)\to e^{\frac {x}{a}} \left (\int _1^x\frac {e^{-\frac {K[1]}{a}} \left (\text {c1} K[1]^k+\text {c2} \log ^n\left (\beta \left (y+\frac {b (K[1]-x)}{a}\right )\right )\right )}{a}dK[1]+c_1\left (y-\frac {b x}{a}\right )\right )\right \}\right \} \]

Maple ✓

 
unassign('w,x,y,a,b,n,m,c,k,alpha,beta,g,A,f,C,lambda,B,mu,d,s,t'); 
unassign('v,q,p,l,g1,g2,g0,h0,h1,h2,f2,f3,c0,c1,c2,a1,a0,b0,b1,b2'); 
unassign('k0,k1,k2,s0,s1,k22,k12,k11,s22,s12,s11,sigma,lambda1,lambda2,n1,n2,nu'); 
pde :=  a*diff(w(x,y),x)+ b*diff(w(x,y),y) = w(x,y)+c1*x^k+c2*ln(beta*y)^n; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime')); 
if(not evalb(sol=())) then sol:=simplify(sol,size); fi;
 

\[ w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac {1}{a}{{\rm e}^{-{\frac {{\it \_a}}{a}}}} \left ( {\it c2}\, \left ( \ln \left ( {\frac {\beta \, \left ( ya-b \left ( x-{\it \_a} \right ) \right ) }{a}} \right ) \right ) ^{n}+{\it c1}\,{{\it \_a}}^{k} \right ) }{d{\it \_a}}+{\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) \right ) {{\rm e}^{{\frac {x}{a}}}} \]

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137.2 Problem 2

problem number 1092

Added April 8, 2019.

Problem Chapter 5.5.2.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b w_y = c w + x^k \ln ^n(\beta y) \]

Mathematica ✓

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; 
 ClearAll[lambda, B, mu, d, g, B, v, f, h, q, p, delta, t]; 
 ClearAll[g1, g0, h2, h1, h0, f1, f2,sigma,lambda1,lambda2,n1,n2]; 
 ClearAll[a1, a0, b2, b1, b0, c2, c1, c0, k0, k1, k2, s1, s0, k22, k11, k12, s11, s22, s12, nu]; 
 pde = a*D[w[x, y], x] + b*D[w[x, y], y] == c*w[x,y]+x^k*Log[beta*y]^n; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; 
 sol = Simplify[sol];
 

\[ \left \{\left \{w(x,y)\to e^{\frac {c x}{a}} \left (\int _1^x\frac {e^{-\frac {c K[1]}{a}} K[1]^k \log ^n\left (\beta \left (y+\frac {b (K[1]-x)}{a}\right )\right )}{a}dK[1]+c_1\left (y-\frac {b x}{a}\right )\right )\right \}\right \} \]

Maple ✓

 
unassign('w,x,y,a,b,n,m,c,k,alpha,beta,g,A,f,C,lambda,B,mu,d,s,t'); 
unassign('v,q,p,l,g1,g2,g0,h0,h1,h2,f2,f3,c0,c1,c2,a1,a0,b0,b1,b2'); 
unassign('k0,k1,k2,s0,s1,k22,k12,k11,s22,s12,s11,sigma,lambda1,lambda2,n1,n2,nu'); 
pde :=  a*diff(w(x,y),x)+ b*diff(w(x,y),y) = c*w(x,y)+x^k*ln(beta*y)^n; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime')); 
if(not evalb(sol=())) then sol:=simplify(sol,size); fi;
 

\[ w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac {{{\it \_a}}^{k}}{a} \left ( \ln \left ( {\frac {\beta \, \left ( ya-b \left ( x-{\it \_a} \right ) \right ) }{a}} \right ) \right ) ^{n}{{\rm e}^{-{\frac {{\it \_a}\,c}{a}}}}}{d{\it \_a}}+{\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) \right ) {{\rm e}^{{\frac {cx}{a}}}} \]

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137.3 Problem 3

problem number 1093

Added April 8, 2019.

Problem Chapter 5.5.2.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a x^k w_x + b x^n w_y = c w + s \ln ^m(\beta x) \]

Mathematica ✓

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; 
 ClearAll[lambda, B, mu, d, g, B, v, f, h, q, p, delta, t]; 
 ClearAll[g1, g0, h2, h1, h0, f1, f2,sigma,lambda1,lambda2,n1,n2]; 
 ClearAll[a1, a0, b2, b1, b0, c2, c1, c0, k0, k1, k2, s1, s0, k22, k11, k12, s11, s22, s12, nu]; 
 pde = a*x^k*D[w[x, y], x] + b*x^n*D[w[x, y], y] == c*w[x,y]+s*Log[beta*x]^m; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; 
 sol = Simplify[sol];
 

\[ \left \{\left \{w(x,y)\to e^{\frac {c x^{1-k}}{a-a k}} \left (\int _1^x\frac {e^{\frac {c K[1]^{1-k}}{a (k-1)}} s K[1]^{-k} \log ^m(\beta K[1])}{a}dK[1]+c_1\left (y-\frac {b x^{-k+n+1}}{a (-k)+a n+a}\right )\right )\right \}\right \} \]

Maple ✓

 
unassign('w,x,y,a,b,n,m,c,k,alpha,beta,g,A,f,C,lambda,B,mu,d,s,t'); 
unassign('v,q,p,l,g1,g2,g0,h0,h1,h2,f2,f3,c0,c1,c2,a1,a0,b0,b1,b2'); 
unassign('k0,k1,k2,s0,s1,k22,k12,k11,s22,s12,s11,sigma,lambda1,lambda2,n1,n2,nu'); 
pde :=  a*x^k*diff(w(x,y),x)+ b*x^n*diff(w(x,y),y) = c*w(x,y)+s*ln(beta*x)^m; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime')); 
if(not evalb(sol=())) then sol:=simplify(sol,size); fi;
 

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

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137.4 Problem 4

problem number 1094

Added April 8, 2019.

Problem Chapter 5.5.2.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a x^n w_x + b y^k w_y = c w + s \ln ^m(\beta x) \]

Mathematica ✓

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; 
 ClearAll[lambda, B, mu, d, g, B, v, f, h, q, p, delta, t]; 
 ClearAll[g1, g0, h2, h1, h0, f1, f2,sigma,lambda1,lambda2,n1,n2]; 
 ClearAll[a1, a0, b2, b1, b0, c2, c1, c0, k0, k1, k2, s1, s0, k22, k11, k12, s11, s22, s12, nu]; 
 pde = a*x^n*D[w[x, y], x] + b*y^k*D[w[x, y], y] == c*w[x,y]+s*Log[beta*x]^m; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; 
 sol = Simplify[sol];
 

\[ \left \{\left \{w(x,y)\to e^{\frac {c x^{1-n}}{a-a n}} \left (\int _1^x\frac {e^{\frac {c K[1]^{1-n}}{a (n-1)}} s K[1]^{-n} \log ^m(\beta K[1])}{a}dK[1]+c_1\left (\frac {b x^{1-n}}{a (n-1)}-\frac {y^{1-k}}{k-1}\right )\right )\right \}\right \} \]

Maple ✓

 
unassign('w,x,y,a,b,n,m,c,k,alpha,beta,g,A,f,C,lambda,B,mu,d,s,t'); 
unassign('v,q,p,l,g1,g2,g0,h0,h1,h2,f2,f3,c0,c1,c2,a1,a0,b0,b1,b2'); 
unassign('k0,k1,k2,s0,s1,k22,k12,k11,s22,s12,s11,sigma,lambda1,lambda2,n1,n2,nu'); 
pde :=  a*x^n*diff(w(x,y),x)+ b*y^k*diff(w(x,y),y) = c*w(x,y)+s*ln(beta*x)^m; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime')); 
if(not evalb(sol=())) then sol:=simplify(sol,size); fi;
 

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

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137.5 Problem 5

problem number 1095

Added April 8, 2019.

Problem Chapter 5.5.2.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a x^n w_x + b \ln ^n(\lambda x) w_y = c w + s x^m \]

Mathematica ✓

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; 
 ClearAll[lambda, B, mu, d, g, B, v, f, h, q, p, delta, t]; 
 ClearAll[g1, g0, h2, h1, h0, f1, f2,sigma,lambda1,lambda2,n1,n2]; 
 ClearAll[a1, a0, b2, b1, b0, c2, c1, c0, k0, k1, k2, s1, s0, k22, k11, k12, s11, s22, s12, nu]; 
 pde = a*x^n*D[w[x, y], x] + b*Log[lambda*x]^n*D[w[x, y], y] == c*w[x,y]+s*x^m; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; 
 sol = Simplify[sol];
 

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

Maple ✓

 
unassign('w,x,y,a,b,n,m,c,k,alpha,beta,g,A,f,C,lambda,B,mu,d,s,t'); 
unassign('v,q,p,l,g1,g2,g0,h0,h1,h2,f2,f3,c0,c1,c2,a1,a0,b0,b1,b2'); 
unassign('k0,k1,k2,s0,s1,k22,k12,k11,s22,s12,s11,sigma,lambda1,lambda2,n1,n2,nu'); 
pde :=  a*x^n*diff(w(x,y),x)+ b*ln(lambda*x)^n*diff(w(x,y),y) = c*w(x,y)+s*x^m; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime')); 
if(not evalb(sol=())) then sol:=simplify(sol,size); fi;
 

\[ w \left ( x,y \right ) ={\frac {1}{ac \left ( m-3\,n+3 \right ) \left ( m-2\,n+2 \right ) \left ( -n+m+1 \right ) }{{\rm e}^{-{\frac {c{x}^{-n+1}}{a \left ( n-1 \right ) }}}} \left ( -a{{\rm e}^{1/2\,{\frac {c{x}^{-n+1}}{a \left ( n-1 \right ) }}}} \left ( -{\frac {c{x}^{-n+1}}{a \left ( n-1 \right ) }} \right ) ^{{\frac {m-2\,n+2}{2\,n-2}}} \left ( -{\frac {c}{a \left ( n-1 \right ) }} \right ) ^{{\frac {n-m-1}{n-1}}} \left ( -{\frac {c}{a \left ( n-1 \right ) }} \right ) ^{{\frac {-n+m+1}{n-1}}}{x}^{m}s \left ( n-1 \right ) \left ( m-2\,n+2 \right ) ^{2} \WhittakerM \left ( {\frac {-m+2\,n-2}{2\,n-2}},{\frac {-m+3\,n-3}{2\,n-2}},-{\frac {c{x}^{-n+1}}{a \left ( n-1 \right ) }} \right ) + \left ( c{x}^{-n+1}+a \left ( m-2\,n+2 \right ) \right ) {{\rm e}^{1/2\,{\frac {c{x}^{-n+1}}{a \left ( n-1 \right ) }}}}{x}^{m}s \left ( -{\frac {c{x}^{-n+1}}{a \left ( n-1 \right ) }} \right ) ^{{\frac {m-2\,n+2}{2\,n-2}}} \left ( -{\frac {c}{a \left ( n-1 \right ) }} \right ) ^{{\frac {n-m-1}{n-1}}} \left ( -{\frac {c}{a \left ( n-1 \right ) }} \right ) ^{{\frac {-n+m+1}{n-1}}} \left ( n-1 \right ) ^{2} \WhittakerM \left ( -{\frac {m}{2\,n-2}},{\frac {-m+3\,n-3}{2\,n-2}},-{\frac {c{x}^{-n+1}}{a \left ( n-1 \right ) }} \right ) +{\it \_F1} \left ( -{\frac {b\int \! \left ( \ln \left ( \lambda \,x \right ) \right ) ^{n}{x}^{-n}\,{\rm d}x}{a}}+y \right ) ac \left ( m-2\,n+2 \right ) \left ( -n+m+1 \right ) \left ( m-3\,n+3 \right ) \right ) } \]

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137.6 Problem 6

problem number 1096

Added April 8, 2019.

Problem Chapter 5.5.2.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a y^k w_x + b x^n w_y = c w + s \ln ^m(\beta x) \]

Mathematica ✓

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; 
 ClearAll[lambda, B, mu, d, g, B, v, f, h, q, p, delta, t]; 
 ClearAll[g1, g0, h2, h1, h0, f1, f2,sigma,lambda1,lambda2,n1,n2]; 
 ClearAll[a1, a0, b2, b1, b0, c2, c1, c0, k0, k1, k2, s1, s0, k22, k11, k12, s11, s22, s12, nu]; 
 pde = a*y^k*D[w[x, y], x] + b*x^n*D[w[x, y], y] == c*w[x,y]+s*Log[beta*x]^m; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; 
 sol = Simplify[sol];
 

\[ \left \{\left \{w(x,y)\to \exp \left (\frac {c x \left (\left (y^{-k-1}\right )^{-\frac {1}{k+1}}\right )^{-k} \left (\frac {a (n+1) y^{k+1}}{a (n+1) y^{k+1}-b (k+1) x^{n+1}}\right )^{\frac {k}{k+1}} \, _2F_1\left (\frac {k}{k+1},\frac {1}{n+1};1+\frac {1}{n+1};\frac {b (k+1) x^{n+1}}{b (k+1) x^{n+1}-a (n+1) y^{k+1}}\right )}{a}\right ) \left (\int _1^x\frac {\exp \left (-\frac {c \, _2F_1\left (\frac {k}{k+1},\frac {1}{n+1};1+\frac {1}{n+1};\frac {b (k+1) K[1]^{n+1}}{b (k+1) x^{n+1}-a (n+1) y^{k+1}}\right ) K[1] \left (1-\frac {b (k+1) K[1]^{n+1}}{b (k+1) x^{n+1}-a (n+1) y^{k+1}}\right )^{\frac {k}{k+1}} \left (\left (\frac {a (n+1)}{a (n+1) y^{k+1}-b (k+1) \left (x^{n+1}-K[1]^{n+1}\right )}\right )^{-\frac {1}{k+1}}\right )^{-k}}{a}\right ) s \left (\left (\frac {a (n+1)}{a (n+1) y^{k+1}-b (k+1) \left (x^{n+1}-K[1]^{n+1}\right )}\right )^{-\frac {1}{k+1}}\right )^{-k} \log ^m(\beta K[1])}{a}dK[1]+c_1\left (\frac {y^{k+1}}{k+1}-\frac {b x^{n+1}}{a n+a}\right )\right )\right \}\right \} \]

Maple ✓

 
unassign('w,x,y,a,b,n,m,c,k,alpha,beta,g,A,f,C,lambda,B,mu,d,s,t'); 
unassign('v,q,p,l,g1,g2,g0,h0,h1,h2,f2,f3,c0,c1,c2,a1,a0,b0,b1,b2'); 
unassign('k0,k1,k2,s0,s1,k22,k12,k11,s22,s12,s11,sigma,lambda1,lambda2,n1,n2,nu'); 
pde :=  a*y^k*diff(w(x,y),x)+ b*x^n*diff(w(x,y),y) = c*w(x,y)+s*ln(beta*x)^m; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime')); 
if(not evalb(sol=())) then sol:=simplify(sol,size); fi;
 

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

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137.7 Problem 7

problem number 1097

Added April 8, 2019.

Problem Chapter 5.5.2.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a y^k w_x + b \ln ^n(\lambda x) w_y = c w + s x^m \]

Mathematica ✓

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; 
 ClearAll[lambda, B, mu, d, g, B, v, f, h, q, p, delta, t]; 
 ClearAll[g1, g0, h2, h1, h0, f1, f2,sigma,lambda1,lambda2,n1,n2]; 
 ClearAll[a1, a0, b2, b1, b0, c2, c1, c0, k0, k1, k2, s1, s0, k22, k11, k12, s11, s22, s12, nu]; 
 pde = a*y^k*D[w[x, y], x] + b*Log[lambda*x]^n*D[w[x, y], y] == c*w[x,y]+s*x^m; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; 
 sol = Simplify[sol];
 

\[ \left \{\left \{w(x,y)\to \exp \left (\int _1^x\frac {c \left (\left (\frac {a \lambda y^{k+1}-b (k+1) \text {Gamma}(n+1,-\log (\lambda x)) (-\log (\lambda x))^{-n} \log ^n(\lambda x)+b (k+1) \text {Gamma}(n+1,-\log (\lambda K[1])) (-\log (\lambda K[1]))^{-n} \log ^n(\lambda K[1])}{a \lambda }\right )^{\frac {1}{k+1}}\right )^{-k}}{a}dK[1]\right ) \left (\int _1^x\frac {\exp \left (-\int _1^{K[2]}\frac {c \left (\left (\frac {a \lambda y^{k+1}-b (k+1) \text {Gamma}(n+1,-\log (\lambda x)) (-\log (\lambda x))^{-n} \log ^n(\lambda x)+b (k+1) \text {Gamma}(n+1,-\log (\lambda K[1])) (-\log (\lambda K[1]))^{-n} \log ^n(\lambda K[1])}{a \lambda }\right )^{\frac {1}{k+1}}\right )^{-k}}{a}dK[1]\right ) s K[2]^m \left (\left (\frac {a \lambda y^{k+1}-b (k+1) \text {Gamma}(n+1,-\log (\lambda x)) (-\log (\lambda x))^{-n} \log ^n(\lambda x)+b (k+1) \text {Gamma}(n+1,-\log (\lambda K[2])) (-\log (\lambda K[2]))^{-n} \log ^n(\lambda K[2])}{a \lambda }\right )^{\frac {1}{k+1}}\right )^{-k}}{a}dK[2]+c_1\left (\frac {y^{k+1}}{k+1}-\frac {b (-\log (\lambda x))^{-n} \log ^n(\lambda x) \text {Gamma}(n+1,-\log (\lambda x))}{a \lambda }\right )\right )\right \}\right \} \]

Maple ✓

 
unassign('w,x,y,a,b,n,m,c,k,alpha,beta,g,A,f,C,lambda,B,mu,d,s,t'); 
unassign('v,q,p,l,g1,g2,g0,h0,h1,h2,f2,f3,c0,c1,c2,a1,a0,b0,b1,b2'); 
unassign('k0,k1,k2,s0,s1,k22,k12,k11,s22,s12,s11,sigma,lambda1,lambda2,n1,n2,nu'); 
pde :=  a*y^k*diff(w(x,y),x)+ b*ln(lambda*x)^n*diff(w(x,y),y) = c*w(x,y)+s*x^m; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime')); 
if(not evalb(sol=())) then sol:=simplify(sol,size); fi;
 

\[ w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac {s{{\it \_f}}^{m}}{a} \left ( \left ( {\frac {b \left ( k+1 \right ) \int \! \left ( \ln \left ( \lambda \,{\it \_f} \right ) \right ) ^{n}\,{\rm d}{\it \_f}-b \left ( k+1 \right ) \int \! \left ( \ln \left ( \lambda \,x \right ) \right ) ^{n}\,{\rm d}x+{y}^{k}ya}{a}} \right ) ^{ \left ( k+1 \right ) ^{-1}} \right ) ^{-k}{{\rm e}^{-{\frac {c \left ( \left ( -b \left ( k+1 \right ) \int \! \left ( \ln \left ( \lambda \,x \right ) \right ) ^{n}\,{\rm d}x+{y}^{k}ya \right ) \left ( \ln \left ( \lambda \,{\it \_a} \right ) \right ) ^{-n}+b \left ( k+1 \right ) {\it \_f} \right ) }{ab} \left ( \left ( \left ( {\frac {-b \left ( k+1 \right ) \int \! \left ( \ln \left ( \lambda \,x \right ) \right ) ^{n}\,{\rm d}x+b \left ( k+1 \right ) {\it \_f}\, \left ( \ln \left ( \lambda \,{\it \_a} \right ) \right ) ^{n}+{y}^{k}ya}{a}} \right ) ^{ \left ( k+1 \right ) ^{-1}} \right ) ^{k} \right ) ^{-1}}}}}{d{\it \_f}}+{\it \_F1} \left ( {\frac {-b \left ( k+1 \right ) \int \! \left ( \ln \left ( \lambda \,x \right ) \right ) ^{n}\,{\rm d}x+{y}^{k}ya}{a}} \right ) \right ) {{\rm e}^{{\frac {c \left ( \left ( -b \left ( k+1 \right ) \int \! \left ( \ln \left ( \lambda \,x \right ) \right ) ^{n}\,{\rm d}x+{y}^{k}ya \right ) \left ( \ln \left ( \lambda \,{\it \_a} \right ) \right ) ^{-n}+bx \left ( k+1 \right ) \right ) }{ab} \left ( \left ( \left ( {\frac {-b \left ( k+1 \right ) \int \! \left ( \ln \left ( \lambda \,x \right ) \right ) ^{n}\,{\rm d}x+bx \left ( k+1 \right ) \left ( \ln \left ( \lambda \,{\it \_a} \right ) \right ) ^{n}+{y}^{k}ya}{a}} \right ) ^{ \left ( k+1 \right ) ^{-1}} \right ) ^{k} \right ) ^{-1}}}} \]