135 HFOPDE, chapter 5.4.5

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135.1 Problem 1

problem number 1079

Added April 4, 2019.

Problem Chapter 5.4.5.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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*Sinh[lambda*x]^k+c2*Cosh[beta*y]^n; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; 
 sol = Simplify[sol];
 

\[ \left \{\left \{w(x,y)\to -\frac {\text {c1} \left (e^{\lambda x}-e^{-\lambda x}\right )^k \left (2-2 e^{2 \lambda x}\right )^{-k} \text {Hypergeometric2F1}\left (-k,-\frac {a k \lambda +1}{2 a \lambda },-\frac {1}{2 a \lambda }-\frac {k}{2}+1,e^{2 \lambda x}\right )}{a k \lambda +1}-\frac {\text {c2} 2^{-n} \left (e^{-\beta y}+e^{\beta y}\right )^n \left (e^{2 \beta y}+1\right )^{-n} \text {Hypergeometric2F1}\left (-n,-\frac {b \beta n+1}{2 b \beta },-\frac {1}{2 b \beta }-\frac {n}{2}+1,-e^{2 \beta y}\right )}{b \beta n+1}+e^{\frac {x}{a}} c_1\left (y-\frac {b x}{a}\right )\right \}\right \} \]

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'); 
pde :=  a*diff(w(x,y),x)+ b*diff(w(x,y),y) = w(x,y)+c1*sinh(lambda*x)^k+c2*cosh(beta*y)^n; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime')); 
sol:=simplify(sol,size);
 

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

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135.2 Problem 2

problem number 1080

Added April 4, 2019.

Problem Chapter 5.4.5.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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

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

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,lambda1,lambda2,n1,n2'); 
pde :=  a*diff(w(x,y),x)+ b*diff(w(x,y),y) = c*w(x,y)+sinh(lambda*x)^k*cosh(beta*y)^n; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime')); 
sol:=simplify(sol,size);
 

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

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135.3 Problem 3

problem number 1081

Added April 4, 2019.

Problem Chapter 5.4.5.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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]+ k*Tanh[lambda*x]+s*coth[mu*y]; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; 
 sol = Simplify[sol];
 

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

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

\[ w \left ( x,y \right ) ={{\rm e}^{{\frac {cx}{a}}}} \left ( {\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) +\int ^{x}\!-{\frac {1}{a} \left ( \left ( k-s \right ) \cosh \left ( {\frac { \left ( -\lambda \,{\it \_a}+\mu \,y \right ) a-b\mu \, \left ( x-{\it \_a} \right ) }{a}} \right ) -\cosh \left ( {\frac { \left ( \lambda \,{\it \_a}+\mu \,y \right ) a-b\mu \, \left ( x-{\it \_a} \right ) }{a}} \right ) \left ( s+k \right ) \right ) {{\rm e}^{-{\frac {{\it \_a}\,c}{a}}}} \left ( \sinh \left ( {\frac { \left ( \lambda \,{\it \_a}+\mu \,y \right ) a-b\mu \, \left ( x-{\it \_a} \right ) }{a}} \right ) +\sinh \left ( {\frac { \left ( -\lambda \,{\it \_a}+\mu \,y \right ) a-b\mu \, \left ( x-{\it \_a} \right ) }{a}} \right ) \right ) ^{-1}}{d{\it \_a}} \right ) \]

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135.4 Problem 4

problem number 1082

Added April 4, 2019.

Problem Chapter 5.4.5.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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

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

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

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

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135.5 Problem 5

problem number 1083

Added April 4, 2019.

Problem Chapter 5.4.5.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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

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

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135.6 Problem 6

problem number 1084

Added April 4, 2019.

Problem Chapter 5.4.5.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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