48 HFOPDE, chapter 2.4.4

48.1 problem number 1
48.2 problem number 2
48.3 problem number 3
48.4 problem number 4

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48.1 problem number 1

problem number 412

Added January 10, 2019.

Problem 2.4.4.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for w(x,y)

wx+acoth(λx)wy=0

Mathematica

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d]; 
 pde = D[w[x, y], x] + a*Coth[lambda*x]*D[w[x, y], y] == 0; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

{{w(x,y)c1(λyalog(sinh(λx))λ)}}

Maple

 
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A'; C:='C';lambda:='lambda';s:='s';B:='B';mu:='mu';d:='d'; 
pde := diff(w(x,y),x)+a*coth(lambda*x)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

w(x,y)=_F1(1/2aln(coth(λx)1)+aln(coth(λx)+1)+2yλλ)

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48.2 problem number 2

problem number 413

Added January 10, 2019.

Problem 2.4.4.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for w(x,y)

wx+acoth(λy)wy=0

Mathematica

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d]; 
 pde = D[w[x, y], x] + a*Coth[lambda*y]*D[w[x, y], y] == 0; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

{{w(x,y)c1(log(cosh(λy))aλxλ)}}

Maple

 
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A'; C:='C';lambda:='lambda';s:='s';B:='B';mu:='mu';d:='d'; 
pde := diff(w(x,y),x)+a*coth(lambda*y)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

w(x,y)=_F1(1/22axλ+ln(coth(yλ)1)+ln(coth(yλ)+1)2ln(coth(yλ))aλ)

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48.3 problem number 3

problem number 414

Added January 10, 2019.

Problem 2.4.4.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for w(x,y)

wx+(y2+aλa(a+λ)coth2(λx))wy=0

Mathematica

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d]; 
 pde = D[w[x, y], x] + (y^2 + a*lambda - a*(a + lambda)*Coth[lambda*x]^2)*D[w[x, y], y] == 0; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

{{w(x,y)c1(λ(ye2λxHypergeometric2F1(2aλ,aλ,1aλ,e2λx)+yHypergeometric2F1(2aλ,aλ,1aλ,e2λx)+ae2λxHypergeometric2F1(2aλ,aλ,1aλ,e2λx)+aHypergeometric2F1(2aλ,aλ,1aλ,e2λx)+2ae2λx(1e2λx)2aλ2a(1e2λx)2aλ)2a(ye2x(a+λ)+ae2x(a+λ)+ye2ax+ae2ax))}}

Maple

 
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A'; C:='C';lambda:='lambda';s:='s';B:='B';mu:='mu';d:='d'; 
pde := diff(w(x,y),x)+(y^2 + a*lambda - a*(a+lambda)*coth(lambda*x)^2)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

w(x,y)=_F1(1(coth(λx)LegendreP(aλ,aλ,coth(λx))a+coth(λx)LegendreP(aλ,aλ,coth(λx))λ+yLegendreP(aλ,aλ,coth(λx))LegendreP(a+λλ,aλ,coth(λx))λ)(coth(λx)LegendreQ(aλ,aλ,coth(λx))a+coth(λx)LegendreQ(aλ,aλ,coth(λx))λ+yLegendreQ(aλ,aλ,coth(λx))λLegendreQ(a+λλ,aλ,coth(λx)))1)

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48.4 problem number 4

problem number 415

Added January 10, 2019.

Problem 2.4.4.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for w(x,y)

wx+(y2+3aλλ2a(a+λ)coth2(λx))wy=0

Mathematica

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d]; 
 pde = D[w[x, y], x] + (y^2 + 3*a*lambda - lambda^2 - a*(a + lambda)*Coth[lambda*x]^2)*D[w[x, y], y] == 0; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple

 
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A'; C:='C';lambda:='lambda';s:='s';B:='B';mu:='mu';d:='d'; 
pde := diff(w(x,y),x)+(y^2 + a*lambda  -lambda^2 - a*(a+lambda)*coth(lambda*x)^2)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

w(x,y)=_F1(1(coth(λx)LegendreP(aλ,a2+λ2λ,coth(λx))a+coth(λx)LegendreP(aλ,a2+λ2λ,coth(λx))λ+LegendreP(a+λλ,a2+λ2λ,coth(λx))a2+λ2LegendreP(a+λλ,a2+λ2λ,coth(λx))aLegendreP(a+λλ,a2+λ2λ,coth(λx))λ+LegendreP(aλ,a2+λ2λ,coth(λx))y)(LegendreQ(aλ,a2+λ2λ,coth(λx))coth(λx)a+LegendreQ(aλ,a2+λ2λ,coth(λx))coth(λx)λ+LegendreQ(a+λλ,a2+λ2λ,coth(λx))a2+λ2LegendreQ(a+λλ,a2+λ2λ,coth(λx))aLegendreQ(a+λλ,a2+λ2λ,coth(λx))λ+LegendreQ(aλ,a2+λ2λ,coth(λx))y)1)