142 HFOPDE, chapter 5.6.5

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142.1 Problem 1

problem number 1125

Added April 11, 2019.

Problem Chapter 5.6.5.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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

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

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*sin(lambda*x)^k+c2*cos(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;
 

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

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142.2 Problem 2

problem number 1126

Added April 11, 2019.

Problem Chapter 5.6.5.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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

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

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)+sin(lambda*x)^k*cos(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;
 

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

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142.3 Problem 3

problem number 1127

Added April 11, 2019.

Problem Chapter 5.6.5.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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

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

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*sin(mu*y)*diff(w(x,y),y) =  c*sin(lambda*x)*w(x,y)+k*cos(nu*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;
 

\[ w \left ( x,y \right ) ={{\rm e}^{-{\frac {c\cos \left ( \lambda \,x \right ) }{a\lambda }}}} \left ( \int \!{\frac {k\cos \left ( \nu \,x \right ) +s}{a}{{\rm e}^{{\frac {c\cos \left ( \lambda \,x \right ) }{a\lambda }}}}}\,{\rm d}x+{\it \_F1} \left ( {\frac {a}{b\mu }\ln \left ( \RootOf \left ( \mu \,y-\arctan \left ( 2\,{{\it \_Z}{{\rm e}^{{\frac {b\mu \,x}{a}}}} \left ( {{\it \_Z}}^{2}{{\rm e}^{2\,{\frac {b\mu \,x}{a}}}}+1 \right ) ^{-1}},-{1 \left ( {{\it \_Z}}^{2}{{\rm e}^{2\,{\frac {b\mu \,x}{a}}}}-1 \right ) \left ( {{\it \_Z}}^{2}{{\rm e}^{2\,{\frac {b\mu \,x}{a}}}}+1 \right ) ^{-1}} \right ) \right ) \right ) } \right ) \right ) \]

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142.4 Problem 4

problem number 1128

Added April 11, 2019.

Problem Chapter 5.6.5.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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

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

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*sin(mu*y)*diff(w(x,y),y) =  c*sin(lambda*x)*w(x,y)+k*tan(nu*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;
 

\[ w \left ( x,y \right ) ={{\rm e}^{-{\frac {c\cos \left ( \lambda \,x \right ) }{a\lambda }}}} \left ( {\it \_F1} \left ( {\frac {a}{b\mu }\ln \left ( \RootOf \left ( \mu \,y-\arctan \left ( 2\,{{\it \_Z}{{\rm e}^{{\frac {b\mu \,x}{a}}}} \left ( {{\it \_Z}}^{2}{{\rm e}^{2\,{\frac {b\mu \,x}{a}}}}+1 \right ) ^{-1}},-{1 \left ( {{\it \_Z}}^{2}{{\rm e}^{2\,{\frac {b\mu \,x}{a}}}}-1 \right ) \left ( {{\it \_Z}}^{2}{{\rm e}^{2\,{\frac {b\mu \,x}{a}}}}+1 \right ) ^{-1}} \right ) \right ) \right ) } \right ) +\int \!{\frac {s\cos \left ( \nu \,x \right ) +k\sin \left ( \nu \,x \right ) }{a\cos \left ( \nu \,x \right ) }{{\rm e}^{{\frac {c\cos \left ( \lambda \,x \right ) }{a\lambda }}}}}\,{\rm d}x \right ) \]

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142.5 Problem 5

problem number 1129

Added April 11, 2019.

Problem Chapter 5.6.5.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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*Tan[mu*y]*D[w[x, y], y] == c*Tan[lambda*x]*w[x,y]+k*Cot[nu*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*diff(w(x,y),x)+ b*tan(mu*y)*diff(w(x,y),y) =  c*tan(lambda*x)*w(x,y)+k*cot(nu*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;
 

\[ w \left ( x,y \right ) = \left ( \cos \left ( \lambda \,x \right ) \right ) ^{-{\frac {c}{a\lambda }}} \left ( \int \!{\frac {k\cos \left ( \nu \,x \right ) +\sin \left ( \nu \,x \right ) s}{\sin \left ( \nu \,x \right ) a} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{{\frac {c}{a\lambda }}}}\,{\rm d}x+{\it \_F1} \left ( {\frac {1}{b\mu } \left ( -b\mu \,x+\ln \left ( {\frac {\tan \left ( \mu \,y \right ) }{\sqrt {1+ \left ( \tan \left ( \mu \,y \right ) \right ) ^{2}}}} \right ) a \right ) } \right ) \right ) \]

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142.6 Problem 6

problem number 1130

Added April 11, 2019.

Problem Chapter 5.6.5.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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*Sin[lambda*x]^n*D[w[x, y], x] + b*Cos[mu*x]^m*D[w[x, y], y] == c*Cos[nu*x]^k*w[x,y]+p*Sin[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*sin(lambda*x)^n*diff(w(x,y),x)+ b*cos(mu*x)^m*diff(w(x,y),y) =  c*cos(nu*x)^k*w(x,y)+p*sin(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;
 

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

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142.7 Problem 7

problem number 1131

Added April 11, 2019.

Problem Chapter 5.6.5.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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*Tan[lambda*x]^n*D[w[x, y], x] + b*Cot[mu*x]^m*D[w[x, y], y] == c*Tan[nu*x]^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*tan(lambda*x)^n*diff(w(x,y),x)+ b*cot(mu*x)^m*diff(w(x,y),y) =  c*tan(nu*x)^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;
 

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