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Added Feb. 7, 2019.
Problem 2.8.5.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + \left ( \lambda \sin (\lambda x) y^2 + f(x) \cos (\lambda x) y-f(x) \right ) w_y = 0 \]
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] + (lambda*Sin[lambda*x]*y^2 + f[x]*Cos[lambda*x]*y - 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)+( lambda*sin(lambda*x)*y^2 + f(x)*cos(lambda*x)*y-f(x))*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( {(\cos \left ( \lambda \,x \right ) y-1) \left ( -\cos \left ( \lambda \,x \right ) y\int \!-\lambda \,{{\rm e}^{\int \!{\frac { \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}\sqrt { \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}}f \left ( x \right ) -2\, \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}\lambda +2\,\lambda }{\sin \left ( \lambda \,x \right ) \cos \left ( \lambda \,x \right ) }}\,{\rm d}x}}\sin \left ( \lambda \,x \right ) \,{\rm d}x+{{\rm e}^{\int \!{\frac { \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}\sqrt { \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}}f \left ( x \right ) -2\, \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}\lambda +2\,\lambda }{\sin \left ( \lambda \,x \right ) \cos \left ( \lambda \,x \right ) }}\,{\rm d}x}}\cos \left ( \lambda \,x \right ) +\int \!-\lambda \,{{\rm e}^{\int \!{\frac { \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}\sqrt { \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}}f \left ( x \right ) -2\, \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}\lambda +2\,\lambda }{\sin \left ( \lambda \,x \right ) \cos \left ( \lambda \,x \right ) }}\,{\rm d}x}}\sin \left ( \lambda \,x \right ) \,{\rm d}x \right ) ^{-1}} \right ) \]
____________________________________________________________________________________
Added Feb. 7, 2019.
Problem 2.8.5.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + \left ( f(x) y^2-a^2 f(x)+a \lambda \sin (\lambda x)+a^2 f(x) \sin ^2(\lambda x) \right ) w_y = 0 \]
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^2*f[x] + a*lambda*Sin[lambda*x] + a^2*f[x]*Sin[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-a^2*f(x)+a*lambda*sin(lambda*x)+a^2*f(x)*sin(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'));
\[ \text { sol=() } \]
____________________________________________________________________________________
Added Feb. 7, 2019.
Problem 2.8.5.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + \left ( f(x) y^2-a^2 f(x)+a \lambda \cos (\lambda x)+a^2 f(x) \cos ^2(\lambda x) \right ) w_y = 0 \]
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^2*f[x] + a*lambda*Cos[lambda*x] + a^2*f[x]*Cos[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-a^2*f(x)+a*lambda*cos(lambda*x)+a^2*f(x)*cos(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'));
\[ \text { sol=() } \]
____________________________________________________________________________________
Added Feb. 7, 2019.
Problem 2.8.5.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + \left ( f(x) y^2-a(a f(x)-\lambda ) \tan ^2(\lambda x)+a \lambda \right ) w_y = 0 \]
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*(a*f[x] - lambda)*Tan[lambda*x]^2 + a*lambda)*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*(a*f(x)-lambda)*tan(lambda*x)^2+a*lambda)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[ \text { sol=() } \]
____________________________________________________________________________________
Added Feb. 7, 2019.
Problem 2.8.5.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + \left ( f(x) y^2-a(a f(x)-\lambda ) \cot ^2(\lambda x)+a \lambda \right ) w_y = 0 \]
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*(a*f[x] - lambda)*Cot[lambda*x]^2 + a*lambda)*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*(a*f(x)-lambda)*cot(lambda*x)^2+a*lambda)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[ \text { sol=() } \]