141 HFOPDE, chapter 5.6.4

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141.1 Problem 1

problem number 1118

Added April 11, 2019.

Problem Chapter 5.6.4.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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

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

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,nu'); 
pde :=  a*diff(w(x,y),x)+ b*diff(w(x,y),y) =c*w(x,y)+k*cot(lambda*x+mu*y); 
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;
 

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

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141.2 Problem 2

problem number 1119

Added April 11, 2019.

Problem Chapter 5.6.4.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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*Cot[lambda*x]^k + c2*Cot[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) = w(x,y)+ c1*cot(lambda*x)^k + c2*cot(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;
 

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

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141.3 Problem 3

problem number 1120

Added April 11, 2019.

Problem Chapter 5.6.4.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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

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

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,nu'); 
pde :=  a*diff(w(x,y),x)+ b*diff(w(x,y),y) = c*w(x,y)+ cot(lambda*x)^k *cot(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;
 

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

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141.4 Problem 4

problem number 1121

Added April 11, 2019.

Problem Chapter 5.6.4.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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

\[ \left \{\left \{w(x,y)\to \sin ^{\frac {c}{a \lambda }}(\lambda x) \left (\int _1^x \frac {k \cot (\nu K[1]) \sin ^{-\frac {c}{a \lambda }}(\lambda K[1])}{a} \, dK[1]+c_1\left (\frac {\log (\sec (\mu y))}{\mu }-\frac {b x}{a}\right )\right )\right \},\left \{w(x,y)\to \sin ^{\frac {c}{a \lambda }}(\lambda x) \left (\int _1^x \frac {k \cot (\nu K[2]) \sin ^{-\frac {c}{a \lambda }}(\lambda K[2])}{a} \, dK[2]+c_1\left (\frac {\log (\sec (\mu y))}{\mu }-\frac {b x}{a}\right )\right )\right \}\right \} \]

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*cot(mu*y)*diff(w(x,y),y) = c*cot(lambda*x)*w(x,y)+ k*cot(nu*x); 
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 ( \sin \left ( \lambda \,x \right ) \right ) ^{{\frac {c}{a\lambda }}} \left ( -\int ^{y}\!{\frac {k}{b} \left ( \sin \left ( {\frac {\lambda }{b\mu } \left ( b\mu \,x+a\ln \left ( \cot \left ( \mu \,y \right ) \right ) -1/2\,a\ln \left ( \left ( \cot \left ( \mu \,y \right ) \right ) ^{2}+1 \right ) +1/2\,a\ln \left ( -2\, \left ( -1+\cos \left ( 2\,\mu \,{\it \_a} \right ) \right ) ^{-1} \right ) -a\ln \left ( {\frac {\cos \left ( \mu \,{\it \_a} \right ) }{\sin \left ( \mu \,{\it \_a} \right ) }} \right ) \right ) } \right ) \right ) ^{-{\frac {c}{a\lambda }}} \left ( \sin \left ( 1/2\,{\frac {1}{b\mu } \left ( -\ln \left ( -2\, \left ( -1+\cos \left ( 2\,\mu \,{\it \_a} \right ) \right ) ^{-1} \right ) a\nu +2\,\ln \left ( {\frac {\cos \left ( \mu \,{\it \_a} \right ) }{\sin \left ( \mu \,{\it \_a} \right ) }} \right ) a\nu +\ln \left ( \left ( \cot \left ( \mu \,y \right ) \right ) ^{2}+1 \right ) a\nu -2\,\ln \left ( \cot \left ( \mu \,y \right ) \right ) a\nu -2\,b\mu \, \left ( -\mu \,{\it \_a}+\nu \,x \right ) \right ) } \right ) -\sin \left ( 1/2\,{\frac {1}{b\mu } \left ( -\ln \left ( -2\, \left ( -1+\cos \left ( 2\,\mu \,{\it \_a} \right ) \right ) ^{-1} \right ) a\nu +2\,\ln \left ( {\frac {\cos \left ( \mu \,{\it \_a} \right ) }{\sin \left ( \mu \,{\it \_a} \right ) }} \right ) a\nu +\ln \left ( \left ( \cot \left ( \mu \,y \right ) \right ) ^{2}+1 \right ) a\nu -2\,\ln \left ( \cot \left ( \mu \,y \right ) \right ) a\nu -2\,b\mu \, \left ( \mu \,{\it \_a}+\nu \,x \right ) \right ) } \right ) \right ) \left ( \sin \left ( 1/2\,{\frac {1}{b\mu } \left ( -\ln \left ( -2\, \left ( -1+\cos \left ( 2\,\mu \,{\it \_a} \right ) \right ) ^{-1} \right ) a\nu +2\,\ln \left ( {\frac {\cos \left ( \mu \,{\it \_a} \right ) }{\sin \left ( \mu \,{\it \_a} \right ) }} \right ) a\nu +\ln \left ( \left ( \cot \left ( \mu \,y \right ) \right ) ^{2}+1 \right ) a\nu -2\,\ln \left ( \cot \left ( \mu \,y \right ) \right ) a\nu -2\,b\mu \, \left ( -\mu \,{\it \_a}+\nu \,x \right ) \right ) } \right ) +\sin \left ( 1/2\,{\frac {1}{b\mu } \left ( -\ln \left ( -2\, \left ( -1+\cos \left ( 2\,\mu \,{\it \_a} \right ) \right ) ^{-1} \right ) a\nu +2\,\ln \left ( {\frac {\cos \left ( \mu \,{\it \_a} \right ) }{\sin \left ( \mu \,{\it \_a} \right ) }} \right ) a\nu +\ln \left ( \left ( \cot \left ( \mu \,y \right ) \right ) ^{2}+1 \right ) a\nu -2\,\ln \left ( \cot \left ( \mu \,y \right ) \right ) a\nu -2\,b\mu \, \left ( \mu \,{\it \_a}+\nu \,x \right ) \right ) } \right ) \right ) ^{-1}}{d{\it \_a}}+{\it \_F1} \left ( 1/2\,{\frac {2\,b\mu \,x-a\ln \left ( \left ( \cot \left ( \mu \,y \right ) \right ) ^{2}+1 \right ) +2\,a\ln \left ( \cot \left ( \mu \,y \right ) \right ) }{b\mu }} \right ) \right ) \]

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141.5 Problem 5

problem number 1122

Added April 11, 2019.

Problem Chapter 5.6.4.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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

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

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*diff(w(x,y),x)+ b*y*diff(w(x,y),y) =c*w(x,y)+k*cot(lambda*x+nu*y); 
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;
 

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

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141.6 Problem 6

problem number 1123

Added April 11, 2019.

Problem Chapter 5.6.4.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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*Cot[lambda*x]^n*D[w[x, y], x] + b*Cot[mu*x]^m*D[w[x, y], y] == c*Cot[nu*x]^k*w[x,y]+p*Cot[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*cot(lambda*x)^n*diff(w(x,y),x)+ b*cot(mu*x)^m*diff(w(x,y),y) =c*cot(nu*x)^k*w(x,y)+p*cot(beta*y)^s; 
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 ) ={{\rm e}^{\int \!{\frac { \left ( \cot \left ( \nu \,x \right ) \right ) ^{k}c \left ( \cot \left ( \lambda \,x \right ) \right ) ^{-n}}{a}}\,{\rm d}x}} \left ( {\it \_F1} \left ( {\frac {1}{a} \left ( ya-b\int \! \left ( {\frac {\cos \left ( \mu \,x \right ) }{\sin \left ( \mu \,x \right ) }} \right ) ^{m} \left ( {\frac {\cos \left ( \lambda \,x \right ) }{\sin \left ( \lambda \,x \right ) }} \right ) ^{-n}\,{\rm d}x \right ) } \right ) +\int ^{x}\!{\frac {p}{a} \left ( {1\cos \left ( {\frac {\beta }{a} \left ( b\int \! \left ( {\frac {\cos \left ( {\it \_f}\,\mu \right ) }{\sin \left ( {\it \_f}\,\mu \right ) }} \right ) ^{m} \left ( {\frac {\cos \left ( \lambda \,{\it \_f} \right ) }{\sin \left ( \lambda \,{\it \_f} \right ) }} \right ) ^{-n}\,{\rm d}{\it \_f}+ya-b\int \! \left ( {\frac {\cos \left ( \mu \,x \right ) }{\sin \left ( \mu \,x \right ) }} \right ) ^{m} \left ( {\frac {\cos \left ( \lambda \,x \right ) }{\sin \left ( \lambda \,x \right ) }} \right ) ^{-n}\,{\rm d}x \right ) } \right ) \left ( \sin \left ( {\frac {\beta }{a} \left ( b\int \! \left ( {\frac {\cos \left ( {\it \_f}\,\mu \right ) }{\sin \left ( {\it \_f}\,\mu \right ) }} \right ) ^{m} \left ( {\frac {\cos \left ( \lambda \,{\it \_f} \right ) }{\sin \left ( \lambda \,{\it \_f} \right ) }} \right ) ^{-n}\,{\rm d}{\it \_f}+ya-b\int \! \left ( {\frac {\cos \left ( \mu \,x \right ) }{\sin \left ( \mu \,x \right ) }} \right ) ^{m} \left ( {\frac {\cos \left ( \lambda \,x \right ) }{\sin \left ( \lambda \,x \right ) }} \right ) ^{-n}\,{\rm d}x \right ) } \right ) \right ) ^{-1}} \right ) ^{s} \left ( {\frac {\cos \left ( \lambda \,{\it \_f} \right ) }{\sin \left ( \lambda \,{\it \_f} \right ) }} \right ) ^{-n}{{\rm e}^{-{\frac {c}{a}\int \! \left ( {\frac {\cos \left ( \nu \,{\it \_f} \right ) }{\sin \left ( \nu \,{\it \_f} \right ) }} \right ) ^{k} \left ( {\frac {\cos \left ( \lambda \,{\it \_f} \right ) }{\sin \left ( \lambda \,{\it \_f} \right ) }} \right ) ^{-n}\,{\rm d}{\it \_f}}}}}{d{\it \_f}} \right ) \]

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141.7 Problem 7

problem number 1124

Added April 11, 2019.

Problem Chapter 5.6.4.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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*Cot[lambda*x]^n*D[w[x, y], x] + b*Cot[mu*x]^m*D[w[x, y], y] == c*Cot[nu*y]^k*w[x,y]+p*Cot[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*cot(lambda*x)^n*diff(w(x,y),x)+ b*cot(mu*x)^m*diff(w(x,y),y) =c*cot(nu*y)^k*w(x,y)+p*cot(beta*x)^s; 
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 ) ={{\rm e}^{\int ^{x}\!{\frac {c \left ( \cot \left ( {\it \_b}\,\lambda \right ) \right ) ^{-n}}{a} \left ( \cot \left ( {\frac {\nu }{a} \left ( -b\int \! \left ( {\frac {\cos \left ( \mu \,x \right ) }{\sin \left ( \mu \,x \right ) }} \right ) ^{m} \left ( {\frac {\cos \left ( \lambda \,x \right ) }{\sin \left ( \lambda \,x \right ) }} \right ) ^{-n}\,{\rm d}x+a \left ( \int \!{\frac {b \left ( \cot \left ( {\it \_b}\,\mu \right ) \right ) ^{m} \left ( \cot \left ( {\it \_b}\,\lambda \right ) \right ) ^{-n}}{a}}\,{\rm d}{\it \_b}+y \right ) \right ) } \right ) \right ) ^{k}}{d{\it \_b}}}} \left ( {\it \_F1} \left ( {\frac {1}{a} \left ( ya-b\int \! \left ( {\frac {\cos \left ( \mu \,x \right ) }{\sin \left ( \mu \,x \right ) }} \right ) ^{m} \left ( {\frac {\cos \left ( \lambda \,x \right ) }{\sin \left ( \lambda \,x \right ) }} \right ) ^{-n}\,{\rm d}x \right ) } \right ) +\int ^{x}\!{\frac {p}{a} \left ( {\frac {\cos \left ( \lambda \,{\it \_f} \right ) }{\sin \left ( \lambda \,{\it \_f} \right ) }} \right ) ^{-n} \left ( {\frac {\cos \left ( \beta \,{\it \_f} \right ) }{\sin \left ( \beta \,{\it \_f} \right ) }} \right ) ^{s}{{\rm e}^{-{\frac {c}{a}\int \! \left ( {1\cos \left ( {\frac {\nu }{a} \left ( b\int \! \left ( {\frac {\cos \left ( {\it \_f}\,\mu \right ) }{\sin \left ( {\it \_f}\,\mu \right ) }} \right ) ^{m} \left ( {\frac {\cos \left ( \lambda \,{\it \_f} \right ) }{\sin \left ( \lambda \,{\it \_f} \right ) }} \right ) ^{-n}\,{\rm d}{\it \_f}+ya-b\int \! \left ( {\frac {\cos \left ( \mu \,x \right ) }{\sin \left ( \mu \,x \right ) }} \right ) ^{m} \left ( {\frac {\cos \left ( \lambda \,x \right ) }{\sin \left ( \lambda \,x \right ) }} \right ) ^{-n}\,{\rm d}x \right ) } \right ) \left ( \sin \left ( {\frac {\nu }{a} \left ( b\int \! \left ( {\frac {\cos \left ( {\it \_f}\,\mu \right ) }{\sin \left ( {\it \_f}\,\mu \right ) }} \right ) ^{m} \left ( {\frac {\cos \left ( \lambda \,{\it \_f} \right ) }{\sin \left ( \lambda \,{\it \_f} \right ) }} \right ) ^{-n}\,{\rm d}{\it \_f}+ya-b\int \! \left ( {\frac {\cos \left ( \mu \,x \right ) }{\sin \left ( \mu \,x \right ) }} \right ) ^{m} \left ( {\frac {\cos \left ( \lambda \,x \right ) }{\sin \left ( \lambda \,x \right ) }} \right ) ^{-n}\,{\rm d}x \right ) } \right ) \right ) ^{-1}} \right ) ^{k} \left ( {\frac {\cos \left ( \lambda \,{\it \_f} \right ) }{\sin \left ( \lambda \,{\it \_f} \right ) }} \right ) ^{-n}\,{\rm d}{\it \_f}}}}}{d{\it \_f}} \right ) \]