133 HFOPDE, chapter 5.4.3

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133.1 Problem 1

problem number 1069

Added April 3, 2019.

Problem Chapter 5.4.3.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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

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

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

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

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133.2 Problem 2

problem number 1070

Added April 3, 2019.

Problem Chapter 5.4.3.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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

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

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133.3 Problem 3

problem number 1071

Added April 3, 2019.

Problem Chapter 5.4.3.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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

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

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133.4 Problem 4

problem number 1072

Added April 3, 2019.

Problem Chapter 5.4.3.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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

\[ w \left ( x,y \right ) =\int ^{x}\!{\frac { \left ( \tanh \left ( {\it \_b}\,\lambda \right ) \right ) ^{-n}}{a} \left ( c\tanh \left ( \nu \,{\it \_b} \right ) w_{{x,y}}+ \left ( {1\sinh \left ( {\frac {\beta }{a} \left ( b\int \! \left ( \tanh \left ( {\it \_b}\,\mu \right ) \right ) ^{m} \left ( \tanh \left ( {\it \_b}\,\lambda \right ) \right ) ^{-n}\,{\rm d}{\it \_b}+ya-b\int \! \left ( {\frac {\sinh \left ( \mu \,x \right ) }{\cosh \left ( \mu \,x \right ) }} \right ) ^{m} \left ( {\frac {\sinh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }} \right ) ^{-n}\,{\rm d}x \right ) } \right ) \left ( \cosh \left ( {\frac {\beta }{a} \left ( b\int \! \left ( \tanh \left ( {\it \_b}\,\mu \right ) \right ) ^{m} \left ( \tanh \left ( {\it \_b}\,\lambda \right ) \right ) ^{-n}\,{\rm d}{\it \_b}+ya-b\int \! \left ( {\frac {\sinh \left ( \mu \,x \right ) }{\cosh \left ( \mu \,x \right ) }} \right ) ^{m} \left ( {\frac {\sinh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }} \right ) ^{-n}\,{\rm d}x \right ) } \right ) \right ) ^{-1}} \right ) ^{s}p \right ) }{d{\it \_b}}+{\it \_F1} \left ( {\frac {1}{a} \left ( ya-b\int \! \left ( {\frac {\sinh \left ( \mu \,x \right ) }{\cosh \left ( \mu \,x \right ) }} \right ) ^{m} \left ( {\frac {\sinh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }} \right ) ^{-n}\,{\rm d}x \right ) } \right ) \]

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133.5 Problem 5

problem number 1073

Added April 3, 2019.

Problem Chapter 5.4.3.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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*Tanh[lambda*x]^n*D[w[x, y], x] + b*Tanh[mu*x]^m*D[w[x, y], y] == c*Tanh[nu*y]*w[x,y]+p*Tanh[beta*x]^s; 
 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'); 
pde :=  a*tanh(lambda*x)^n*diff(w(x,y),x)+ b*tanh(mu*x)^m*diff(w(x,y),y) = c*tanh(nu*y)*w(x,y)+p*tanh(beta*x)^s; 
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 ) ={{\rm e}^{\int ^{x}\!{\frac {c \left ( \tanh \left ( {\it \_b}\,\lambda \right ) \right ) ^{-n}}{a}\tanh \left ( {\frac {\nu }{a} \left ( -b\int \! \left ( {\frac {\sinh \left ( \mu \,x \right ) }{\cosh \left ( \mu \,x \right ) }} \right ) ^{m} \left ( {\frac {\sinh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }} \right ) ^{-n}\,{\rm d}x+a \left ( \int \!{\frac {b \left ( \tanh \left ( {\it \_b}\,\mu \right ) \right ) ^{m} \left ( \tanh \left ( {\it \_b}\,\lambda \right ) \right ) ^{-n}}{a}}\,{\rm d}{\it \_b}+y \right ) \right ) } \right ) }{d{\it \_b}}}} \left ( {\it \_F1} \left ( {\frac {1}{a} \left ( ya-b\int \! \left ( {\frac {\sinh \left ( \mu \,x \right ) }{\cosh \left ( \mu \,x \right ) }} \right ) ^{m} \left ( {\frac {\sinh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }} \right ) ^{-n}\,{\rm d}x \right ) } \right ) +\int ^{x}\!{\frac {p}{a} \left ( {\frac {\sinh \left ( \beta \,{\it \_f} \right ) }{\cosh \left ( \beta \,{\it \_f} \right ) }} \right ) ^{s} \left ( {\frac {\sinh \left ( \lambda \,{\it \_f} \right ) }{\cosh \left ( \lambda \,{\it \_f} \right ) }} \right ) ^{-n}{{\rm e}^{-{\frac {c}{a}\int \!{1\sinh \left ( {\frac {\nu }{a} \left ( b\int \! \left ( {\frac {\sinh \left ( {\it \_f}\,\mu \right ) }{\cosh \left ( {\it \_f}\,\mu \right ) }} \right ) ^{m} \left ( {\frac {\sinh \left ( \lambda \,{\it \_f} \right ) }{\cosh \left ( \lambda \,{\it \_f} \right ) }} \right ) ^{-n}\,{\rm d}{\it \_f}+ya-b\int \! \left ( {\frac {\sinh \left ( \mu \,x \right ) }{\cosh \left ( \mu \,x \right ) }} \right ) ^{m} \left ( {\frac {\sinh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }} \right ) ^{-n}\,{\rm d}x \right ) } \right ) \left ( {\frac {\sinh \left ( \lambda \,{\it \_f} \right ) }{\cosh \left ( \lambda \,{\it \_f} \right ) }} \right ) ^{-n} \left ( \cosh \left ( {\frac {\nu }{a} \left ( b\int \! \left ( {\frac {\sinh \left ( {\it \_f}\,\mu \right ) }{\cosh \left ( {\it \_f}\,\mu \right ) }} \right ) ^{m} \left ( {\frac {\sinh \left ( \lambda \,{\it \_f} \right ) }{\cosh \left ( \lambda \,{\it \_f} \right ) }} \right ) ^{-n}\,{\rm d}{\it \_f}+ya-b\int \! \left ( {\frac {\sinh \left ( \mu \,x \right ) }{\cosh \left ( \mu \,x \right ) }} \right ) ^{m} \left ( {\frac {\sinh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }} \right ) ^{-n}\,{\rm d}x \right ) } \right ) \right ) ^{-1}}\,{\rm d}{\it \_f}}}}}{d{\it \_f}} \right ) \]