62 HFOPDE, chapter 2.8.2

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62.1 problem number 1

problem number 573

Added Feb. 4, 2019.

Problem 2.8.2.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( a e^{\lambda x} y^2 + a e^{\lambda x} f(x) y+\lambda f(x) \right ) w_y = 0 \]

Mathematica ✗

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

\[ \text {Failed} \]

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';f:='f'; 
C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v'; 
pde := diff(w(x,y),x)+( a*exp(lambda*x)*y^2 + a*exp(lambda*x)*f(x)*y+lambda*f(x))*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac {\lambda \,{{\rm e}^{\lambda \,x}} \left ( {{\rm e}^{\lambda \,x}}ay+\lambda \right ) }{{{\rm e}^{2\,\lambda \,x}}y\int \!{{\rm e}^{a\int \!{{\rm e}^{\lambda \,x}}f \left ( x \right ) \,{\rm d}x-\lambda \,x}}\,{\rm d}xa+{{\rm e}^{\lambda \,x}}\lambda \,\int \!{{\rm e}^{a\int \!{{\rm e}^{\lambda \,x}}f \left ( x \right ) \,{\rm d}x-\lambda \,x}}\,{\rm d}x+{{\rm e}^{a\int \!{{\rm e}^{\lambda \,x}}f \left ( x \right ) \,{\rm d}x}}}} \right ) \]

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62.2 problem number 2

problem number 574

Added Feb. 4, 2019.

Problem 2.8.2.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( f(x) y^2-a e^{\lambda x} f(x) y+a \lambda e^{\lambda x}\right ) w_y = 0 \]

Mathematica ✗

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

\[ \text {Failed} \]

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';f:='f'; 
C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v'; 
pde := diff(w(x,y),x)+( f(x)*y^2-a*exp(lambda*x)*f(x)*y+a*lambda*exp(lambda*x))*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[ \text { sol=() } \]

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62.3 problem number 3

problem number 575

Added Feb. 4, 2019.

Problem 2.8.2.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( f(x) y^2+a \lambda e^{\lambda x}-a^2 e^{2 \lambda x} f(x)\right ) w_y = 0 \]

Mathematica ✗

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

\[ \text {Failed} \]

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';f:='f'; 
C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v'; 
pde := diff(w(x,y),x)+( f(x)*y^2+a*lambda*exp(lambda*x)-a^2*exp(2*lambda*x)*f(x))*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[ \text { sol=() } \]

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62.4 problem number 4

problem number 576

Added Feb. 4, 2019.

Problem 2.8.2.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( f(x) y^2+\lambda y+ a e^{2 \lambda x} f(x)\right ) w_y = 0 \]

Mathematica ✓

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

\[ \left \{\left \{w(x,y)\to c_1\left (\tan ^{-1}\left (\frac {y e^{-\lambda x}}{\sqrt {a}}\right )-\sqrt {a} \int _1^x f(K[1]) e^{\lambda K[1]} \, dK[1]\right )\right \}\right \} \]

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';f:='f'; 
C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v'; 
pde := diff(w(x,y),x)+(  f(x)*y^2+lambda*y+ a*exp(2*lambda*x)* f(x))*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) assuming a>0),output='realtime'));
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( \sqrt {a}\int \!{{\rm e}^{\lambda \,x}}f \left ( x \right ) \,{\rm d}x-\arctan \left ( {\frac {{{\rm e}^{-\lambda \,x}}y}{\sqrt {a}}} \right ) \right ) \]

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62.5 problem number 5

problem number 577

Added Feb. 4, 2019.

Problem 2.8.2.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( f(x) y^2-(a e^{\lambda x}+b) f(x) y+a \lambda e^{\lambda x}\right ) w_y = 0 \]

Mathematica ✗

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

\[ \text {Failed} \]

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';f:='f'; 
C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v'; 
pde := diff(w(x,y),x)+(  f(x)*y^2-(a*exp(lambda*x)+b)*f(x)*y+a *lambda*exp(lambda*x))*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

\[ \text { sol=() } \]

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62.6 problem number 6

problem number 578

Added Feb. 4, 2019.

Problem 2.8.2.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left (e^{\lambda x} f(x) y^2+(a f(x)-\lambda ) y+b e^{-\lambda x} f(x)\right ) w_y = 0 \]

Mathematica ✗

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

\[ \text {Failed} \]

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';f:='f'; 
C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v'; 
pde := diff(w(x,y),x)+(  exp(lambda*x)*f(x)*y^2+(a*f(x)-lambda)*y+b*exp(-lambda*x)*f(x))*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {a}{\sqrt {{a}^{2} \left ( {a}^{2}-4\,b \right ) }} \left ( -\sqrt {{a}^{2} \left ( {a}^{2}-4\,b \right ) }\int \!f \left ( x \right ) \,{\rm d}x-2\,a\arctanh \left ( {\frac {a \left ( 2\,y{{\rm e}^{\lambda \,x}}+a \right ) }{\sqrt {{a}^{2} \left ( {a}^{2}-4\,b \right ) }}} \right ) \right ) } \right ) \]

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62.7 problem number 7

problem number 579

Added Feb. 4, 2019.

Problem 2.8.2.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left (f(x) y^2+ g(x) y+a \lambda e^{\lambda x} -a e^{\lambda x} g(x) -a^2 e^{2 \lambda x} f(x)\right ) w_y = 0 \]

Mathematica ✗

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

\[ \text {Failed} \]

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';f:='f'; 
C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v'; 
pde := diff(w(x,y),x)+(  f(x)*y^2+ g(x)*y+a*lambda*exp(lambda*x) -a*exp(lambda*x)*g(x)-a^2*exp(2*lambda*x)*f(x))*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

\[ \text { sol=() } \]

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62.8 problem number 8

problem number 580

Added Feb. 7, 2019.

Problem 2.8.2.8 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left (f(x) y^2- a e^{\lambda x} g(x) y + a \lambda e^{\lambda x} +a^2 e^{2 \lambda x} (g(x)-f(x))\right ) w_y = 0 \]

Mathematica ✗

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

\[ \text {Failed} \]

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';f:='f'; 
C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v'; 
pde := diff(w(x,y),x)+(  f(x)*y^2- a*exp(lambda*x)*g(x)*y + a*lambda*exp(lambda*x) +a^2*exp(2*lambda*x)* (g(x)-f(x)))*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

\[ \text { sol=() } \]

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62.9 problem number 9

problem number 581

Added Feb. 7, 2019.

Problem 2.8.2.9 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left (f(x) y^2+2 a \lambda x e^{\lambda x^2} - a^2 f(x) e^{2 \lambda x^2} \right ) w_y = 0 \]

Mathematica ✗

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

\[ \text {Failed} \]

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';f:='f'; 
C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v'; 
pde := diff(w(x,y),x)+(  f(x)*y^2+2*a*lambda*x*exp(lambda*x^2) - a^2*f(x)*exp(2*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'));
 

\[ \text { sol=() } \]

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62.10 problem number 10

problem number 582

Added Feb. 7, 2019.

Problem 2.8.2.10 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left (f(x) y^2+2 \lambda x y+ a f(x) e^{2 \lambda x^2} \right ) w_y = 0 \]

Mathematica ✓

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

\[ \left \{\left \{w(x,y)\to c_1\left (\tan ^{-1}\left (\frac {y e^{-\lambda x^2}}{\sqrt {a}}\right )-\sqrt {a} \int _1^x f(K[1]) e^{\lambda K[1]^2} \, dK[1]\right )\right \}\right \} \]

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';f:='f'; 
C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v'; 
pde := diff(w(x,y),x)+(  f(x)*y^2+2*lambda*x*y+ a*f(x)*exp(2*lambda*x^2))*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) assuming a>0 ),output='realtime'));
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( \sqrt {a}\int \!{{\rm e}^{\lambda \,{x}^{2}}}f \left ( x \right ) \,{\rm d}x-\arctan \left ( {\frac {{{\rm e}^{-\lambda \,{x}^{2}}}y}{\sqrt {a}}} \right ) \right ) \]

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62.11 problem number 11

problem number 583

Added Feb. 7, 2019.

Problem 2.8.2.10 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left (f(x) e^{\lambda y} + g(x) \right ) w_y = 0 \]

Mathematica ✗

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

\[ \text {Failed} \]

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';f:='f'; 
C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v'; 
pde := diff(w(x,y),x)+(   f(x)*exp(lambda*y) + g(x))*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac {\lambda \,\int \!f \left ( x \right ) {{\rm e}^{\lambda \,\int \!g \left ( x \right ) \,{\rm d}x}}\,{\rm d}x+{{\rm e}^{\lambda \, \left ( \int \!g \left ( x \right ) \,{\rm d}x-y \right ) }}}{\lambda }} \right ) \]