136 HFOPDE, chapter 5.5.1

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136.1 Problem 1

problem number 1085

Added April 5, 2019.

Problem Chapter 5.5.1.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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]+ Log[lambda*x]^k*Log[beta*y]^n; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; 
 sol = Simplify[sol];
 

\[ \text {\$Aborted} \]

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)+ln(lambda*x)^k*ln(beta*y)^n; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime')); 
sol:=simplify(sol,size);
 

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

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136.2 Problem 2

problem number 1086

Added April 5, 2019.

Problem Chapter 5.5.1.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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*D[w[x, y], x] + b*D[w[x, y], y] == c*Log[lambda*x]^k*w[x,y]+s*Log[beta*x]^n; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; 
 sol = Simplify[sol];
 

\[ \text {\$Aborted} \]

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*ln(lambda*x)^k*w(x,y)+s*ln(beta*x)^n; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime')); 
sol:=simplify(sol,size);
 

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

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136.3 Problem 3

problem number 1087

Added April 5, 2019.

Problem Chapter 5.5.1.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b w_y = \left ( c_1 \ln ^{n_1}(\lambda _1 x) +c_2 \ln ^{n_2}(\lambda _2 y) \right ) w + s_1 \ln ^{k_1}(\beta _1 x)+s_2 \ln ^{k_2}(\beta _2 y) \]

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] == ( c1*Log[lambda1*x]^n1 +c2*Log[lambda2*y]^n2)*w[x,y] + s1*Log[beta1*x]^k1+s2*Log[beta2*y]*k2; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; 
 sol = Simplify[sol];
 

\[ \text {\$Aborted} \]

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) = ( c1*ln(lambda1*x)^n1 +c2*ln(lambda2*y)^n2)*w(x,y) + s1*ln(beta1*x)^k1+s2*ln(beta2*y)*k2; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime')); 
sol:=simplify(sol,size);
 

\[ w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac {1}{a}{{\rm e}^{{\frac {1}{a} \left ( -{\it c1}\,\int \! \left ( \ln \left ( \lambda 1\,{\it \_f} \right ) \right ) ^{{\it n1}}\,{\rm d}{\it \_f}-{\it c2}\,\int \! \left ( \ln \left ( {\frac {\lambda 2\, \left ( ya-b \left ( x-{\it \_f} \right ) \right ) }{a}} \right ) \right ) ^{{\it n2}}\,{\rm d}{\it \_f} \right ) }}} \left ( {\it s2}\,\ln \left ( {\frac {\beta 2\, \left ( ya-b \left ( x-{\it \_f} \right ) \right ) }{a}} \right ) {\it k2}+{\it s1}\, \left ( \ln \left ( \beta 1\,{\it \_f} \right ) \right ) ^{{\it k1}} \right ) }{d{\it \_f}}+{\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) \right ) {{\rm e}^{\int ^{x}\!{\frac {1}{a} \left ( {\it c1}\, \left ( \ln \left ( \lambda 1\,{\it \_a} \right ) \right ) ^{{\it n1}}+ \left ( \ln \left ( {\frac {\lambda 2\, \left ( ya-b \left ( x-{\it \_a} \right ) \right ) }{a}} \right ) \right ) ^{{\it n2}}{\it c2} \right ) }{d{\it \_a}}}} \]

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136.4 Problem 4

problem number 1088

Added April 5, 2019.

Problem Chapter 5.5.1.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a \ln (\lambda x) w_x + b \ln (\mu y) w_y = c w + k \]

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*Log[lambda*x]*D[w[x, y], x] + b*Log[mu*y]*D[w[x, y], y] == c*w[x,y]+k; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; 
 sol = Simplify[sol];
 

\[ \text {Failed} \]

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*ln(lambda*x)*diff(w(x,y),x)+ b*ln(mu*y)*diff(w(x,y),y) =c*w(x,y)+k; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime')); 
sol:=simplify(sol,size);
 

\[ w \left ( x,y \right ) ={\frac {1}{c} \left ( {{\rm e}^{-{\frac {c\Ei \left ( 1,-\ln \left ( \lambda \,x \right ) \right ) }{a\lambda }}}}{\it \_F1} \left ( {\frac {-a\Ei \left ( 1,-\ln \left ( \mu \,y \right ) \right ) \lambda +\Ei \left ( 1,-\ln \left ( \lambda \,x \right ) \right ) b\mu }{\lambda \,b\mu }} \right ) c-k \right ) } \]

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136.5 Problem 5

problem number 1089

Added April 5, 2019.

Problem Chapter 5.5.1.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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*Log[lambda*x]^n*D[w[x, y], x] + b*Log[mu*x]^m*D[w[x, y], y] == c*Log[nu*x]^k*w[x,y]+p*Log[beta*y]^s+q; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; 
 sol = Simplify[sol];
 

\[ \text {\$Aborted} \]

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*ln(lambda*x)^n*diff(w(x,y),x)+ b*ln(mu*x)^m*diff(w(x,y),y) = c*ln(nu*x)^k*w(x,y)+p*ln(beta*y)^s+q; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime')); 
sol:=simplify(sol,size);
 

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

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136.6 Problem 6

problem number 1090

Added April 5, 2019.

Problem Chapter 5.5.1.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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*Log[lambda*x]^n*D[w[x, y], x] + b*Log[mu*x]^m*D[w[x, y], y] == c*Log[nu*y]^k*w[x,y]+p*Log[beta*x]^s+q; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; 
 sol = Simplify[sol];
 

\[ \text {\$Aborted} \]

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*ln(lambda*x)^n*diff(w(x,y),x)+ b*ln(mu*x)^m*diff(w(x,y),y) = c*ln(nu*y)^k*w(x,y)+p*ln(beta*x)^s+q; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime')); 
sol:=simplify(sol,size);
 

\[ w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac { \left ( \ln \left ( \lambda \,{\it \_f} \right ) \right ) ^{-n} \left ( p \left ( \ln \left ( \beta \,{\it \_f} \right ) \right ) ^{s}+q \right ) }{a}{{\rm e}^{-{\frac {c}{a}\int \! \left ( \ln \left ( {\frac {\nu \, \left ( ya-b\int \! \left ( \ln \left ( \lambda \,x \right ) \right ) ^{-n} \left ( \ln \left ( \mu \,x \right ) \right ) ^{m}\,{\rm d}x+b\int \! \left ( \ln \left ( \lambda \,{\it \_f} \right ) \right ) ^{-n} \left ( \ln \left ( {\it \_f}\,\mu \right ) \right ) ^{m}\,{\rm d}{\it \_f} \right ) }{a}} \right ) \right ) ^{k} \left ( \ln \left ( \lambda \,{\it \_f} \right ) \right ) ^{-n}\,{\rm d}{\it \_f}}}}}{d{\it \_f}}+{\it \_F1} \left ( -{\frac {b\int \! \left ( \ln \left ( \lambda \,x \right ) \right ) ^{-n} \left ( \ln \left ( \mu \,x \right ) \right ) ^{m}\,{\rm d}x}{a}}+y \right ) \right ) {{\rm e}^{\int ^{x}\!{\frac {c \left ( \ln \left ( {\it \_b}\,\lambda \right ) \right ) ^{-n}}{a} \left ( \ln \left ( \nu \, \left ( \int \!{\frac {b \left ( \ln \left ( {\it \_b}\,\mu \right ) \right ) ^{m} \left ( \ln \left ( {\it \_b}\,\lambda \right ) \right ) ^{-n}}{a}}\,{\rm d}{\it \_b}-{\frac {b\int \! \left ( \ln \left ( \lambda \,x \right ) \right ) ^{-n} \left ( \ln \left ( \mu \,x \right ) \right ) ^{m}\,{\rm d}x}{a}}+y \right ) \right ) \right ) ^{k}}{d{\it \_b}}}} \]