____________________________________________________________________________________
Added January 14, 2019.
Problem 2.6.3.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + \left ( a \tan ^k(\lambda x)+b\right ) w_y = 0 \]
Mathematica ✓
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B]; pde = D[w[x, y], x] + (a + Tan[lambda*x] + b)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to c_1\left (\frac {-a \lambda x-b \lambda x+\log (\cos (\lambda x))+\lambda y}{\lambda }\right )\right \}\right \} \]
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'; C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s'; pde := diff(w(x,y),x)+(a+tan(lambda*x)+b)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( 1/2\,{\frac {-2\,ax\lambda -2\,bx\lambda +2\,y\lambda -\ln \left ( 1+ \left ( \tan \left ( \lambda \,x \right ) \right ) ^{2} \right ) }{\lambda }} \right ) \]
____________________________________________________________________________________
Added January 14, 2019.
Problem 2.6.3.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + \left ( a \tan ^k(\lambda y)+b\right ) w_y = 0 \]
Mathematica ✓
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B]; pde = D[w[x, y], x] + (a + Tan[lambda*y] + b)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to c_1\left (-x+\frac {2 (a+b) \tan ^{-1}(\tan (\lambda y))+2 \log (a+b+\tan (\lambda y))-\log \left (\sec ^2(\lambda y)\right )}{2 \lambda (a+b-i) (a+b+i)}\right )\right \}\right \} \]
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'; C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s'; pde := diff(w(x,y),x)+(a+tan(lambda*y)+b)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( 1/2\,{\frac {2\,{a}^{2}\lambda \,x+4\,ab\lambda \,x+2\,{b}^{2}\lambda \,x+2\,\lambda \,x-2\,\arctan \left ( \tan \left ( y\lambda \right ) \right ) a-2\,\arctan \left ( \tan \left ( y\lambda \right ) \right ) b-2\,\ln \left ( a+\tan \left ( y\lambda \right ) +b \right ) +\ln \left ( 1+ \left ( \tan \left ( y\lambda \right ) \right ) ^{2} \right ) }{\lambda \, \left ( {a}^{2}+2\,ab+{b}^{2}+1 \right ) }} \right ) \]
____________________________________________________________________________________
Added January 14, 2019.
Problem 2.6.3.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + a \tan ^k(\lambda x) \tan ^n(\mu y) w_y = 0 \]
Mathematica ✓
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B]; pde = D[w[x, y], x] + a*Tan[lambda*x]^k*Tan[mu*y]^n*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to c_1\left (\frac {-a \tan ^{k+1}(\lambda x) \text {Hypergeometric2F1}\left (1,\frac {k+1}{2},\frac {k+3}{2},-\tan ^2(\lambda x)\right )+\frac {k \lambda \tan ^{1-n}(\mu y) \text {Hypergeometric2F1}\left (1,\frac {1}{2}-\frac {n}{2},\frac {3}{2}-\frac {n}{2},-\tan ^2(\mu y)\right )}{\mu -\mu n}+\frac {\lambda \tan ^{1-n}(\mu y) \text {Hypergeometric2F1}\left (1,\frac {1}{2}-\frac {n}{2},\frac {3}{2}-\frac {n}{2},-\tan ^2(\mu y)\right )}{\mu -\mu n}}{k \lambda +\lambda }\right )\right \}\right \} \]
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'; C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s'; pde := diff(w(x,y),x)+a *tan(lambda*x)^k * tan(mu*y)^n*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( -\int \! \left ( \tan \left ( \lambda \,x \right ) \right ) ^{k}\,{\rm d}x+\int \!{\frac { \left ( \tan \left ( \mu \,y \right ) \right ) ^{-n}}{a}}\,{\rm d}y \right ) \] Has unresolved integrals
____________________________________________________________________________________
Added January 14, 2019.
Problem 2.6.3.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x +\left ( y^2+ a \lambda + a(\lambda -a) \tan ^2(\lambda x)\right ) w_y = 0 \]
Mathematica ✗
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B]; pde = D[w[x, y], x] + (y^2 + a*lambda + a*(lambda - a)*Tan[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'; C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s'; pde := diff(w(x,y),x)+( y^2+ a *lambda + a*(lambda -a) *tan(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'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{1 \left ( 4\,\LegendreP \left ( 1/2\,{\frac {2\,a-\lambda }{\lambda }},1/2\,{\frac {2\,a-\lambda }{\lambda }},\sin \left ( \lambda \,x \right ) \right ) \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}y+\sin \left ( \lambda \,x \right ) \LegendreP \left ( 1/2\,{\frac {2\,a-\lambda }{\lambda }},1/2\,{\frac {2\,a-\lambda }{\lambda }},\sin \left ( \lambda \,x \right ) \right ) a+\LegendreP \left ( 1/2\,{\frac {2\,a-\lambda }{\lambda }},1/2\,{\frac {2\,a-\lambda }{\lambda }},\sin \left ( \lambda \,x \right ) \right ) a\sin \left ( 3\,\lambda \,x \right ) -2\,\LegendreP \left ( 1/2\,{\frac {2\,a+\lambda }{\lambda }},1/2\,{\frac {2\,a-\lambda }{\lambda }},\sin \left ( \lambda \,x \right ) \right ) \lambda \,\cos \left ( 2\,\lambda \,x \right ) -2\,\LegendreP \left ( 1/2\,{\frac {2\,a+\lambda }{\lambda }},1/2\,{\frac {2\,a-\lambda }{\lambda }},\sin \left ( \lambda \,x \right ) \right ) \lambda \right ) \left ( 4\,\LegendreQ \left ( 1/2\,{\frac {2\,a-\lambda }{\lambda }},1/2\,{\frac {2\,a-\lambda }{\lambda }},\sin \left ( \lambda \,x \right ) \right ) \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}y+\LegendreQ \left ( 1/2\,{\frac {2\,a-\lambda }{\lambda }},1/2\,{\frac {2\,a-\lambda }{\lambda }},\sin \left ( \lambda \,x \right ) \right ) \sin \left ( \lambda \,x \right ) a+\LegendreQ \left ( 1/2\,{\frac {2\,a-\lambda }{\lambda }},1/2\,{\frac {2\,a-\lambda }{\lambda }},\sin \left ( \lambda \,x \right ) \right ) \sin \left ( 3\,\lambda \,x \right ) a-2\,\LegendreQ \left ( 1/2\,{\frac {2\,a+\lambda }{\lambda }},1/2\,{\frac {2\,a-\lambda }{\lambda }},\sin \left ( \lambda \,x \right ) \right ) \cos \left ( 2\,\lambda \,x \right ) \lambda -2\,\LegendreQ \left ( 1/2\,{\frac {2\,a+\lambda }{\lambda }},1/2\,{\frac {2\,a-\lambda }{\lambda }},\sin \left ( \lambda \,x \right ) \right ) \lambda \right ) ^{-1}} \right ) \]
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Added January 14, 2019.
Problem 2.6.3.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x +\left ( y^2+ \lambda ^2 +3 a \lambda +a(\lambda -a) \tan ^2(\lambda x)\right ) w_y = 0 \]
Mathematica ✗
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B]; pde = D[w[x, y], x] + (y^2 + lambda^2 + 3*a*lambda + a*(lambda - a)*Tan[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'; C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s'; pde := diff(w(x,y),x)+( y^2+ lambda^2 +3*a*lambda +a*(lambda-a)*tan(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 \left ( x,y \right ) ={\it \_F1} \left ( -{1 \left ( \LegendreP \left ( 1/2\,{\frac {2\,a+\lambda }{\lambda }},1/2\,{\frac {2\,a-\lambda }{\lambda }},\sin \left ( \lambda \,x \right ) \right ) \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3}\lambda -2\,\LegendreP \left ( 1/2\,{\frac {2\,a+\lambda }{\lambda }},1/2\,{\frac {2\,a-\lambda }{\lambda }},\sin \left ( \lambda \,x \right ) \right ) \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}\cos \left ( \lambda \,x \right ) y+2\, \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}\LegendreP \left ( 1/2\,{\frac {2\,a+\lambda }{\lambda }},1/2\,{\frac {2\,a-\lambda }{\lambda }},\sin \left ( \lambda \,x \right ) \right ) \sin \left ( \lambda \,x \right ) a+3\, \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}\LegendreP \left ( 1/2\,{\frac {2\,a+\lambda }{\lambda }},1/2\,{\frac {2\,a-\lambda }{\lambda }},\sin \left ( \lambda \,x \right ) \right ) \sin \left ( \lambda \,x \right ) \lambda -4\, \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}\LegendreP \left ( 1/2\,{\frac {2\,a+3\,\lambda }{\lambda }},1/2\,{\frac {2\,a-\lambda }{\lambda }},\sin \left ( \lambda \,x \right ) \right ) \lambda -\LegendreP \left ( 1/2\,{\frac {2\,a+\lambda }{\lambda }},1/2\,{\frac {2\,a-\lambda }{\lambda }},\sin \left ( \lambda \,x \right ) \right ) \sin \left ( \lambda \,x \right ) \lambda +2\,\LegendreP \left ( 1/2\,{\frac {2\,a+\lambda }{\lambda }},1/2\,{\frac {2\,a-\lambda }{\lambda }},\sin \left ( \lambda \,x \right ) \right ) \cos \left ( \lambda \,x \right ) y \right ) \left ( \LegendreQ \left ( 1/2\,{\frac {2\,a+\lambda }{\lambda }},1/2\,{\frac {2\,a-\lambda }{\lambda }},\sin \left ( \lambda \,x \right ) \right ) \left ( \sin \left ( \lambda \,x \right ) \right ) ^{3}\lambda -2\,\LegendreQ \left ( 1/2\,{\frac {2\,a+\lambda }{\lambda }},1/2\,{\frac {2\,a-\lambda }{\lambda }},\sin \left ( \lambda \,x \right ) \right ) \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}\cos \left ( \lambda \,x \right ) y+2\,\LegendreQ \left ( 1/2\,{\frac {2\,a+\lambda }{\lambda }},1/2\,{\frac {2\,a-\lambda }{\lambda }},\sin \left ( \lambda \,x \right ) \right ) \sin \left ( \lambda \,x \right ) \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}a+3\,\LegendreQ \left ( 1/2\,{\frac {2\,a+\lambda }{\lambda }},1/2\,{\frac {2\,a-\lambda }{\lambda }},\sin \left ( \lambda \,x \right ) \right ) \sin \left ( \lambda \,x \right ) \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}\lambda -4\, \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}\LegendreQ \left ( 1/2\,{\frac {2\,a+3\,\lambda }{\lambda }},1/2\,{\frac {2\,a-\lambda }{\lambda }},\sin \left ( \lambda \,x \right ) \right ) \lambda -\LegendreQ \left ( 1/2\,{\frac {2\,a+\lambda }{\lambda }},1/2\,{\frac {2\,a-\lambda }{\lambda }},\sin \left ( \lambda \,x \right ) \right ) \sin \left ( \lambda \,x \right ) \lambda +2\,\LegendreQ \left ( 1/2\,{\frac {2\,a+\lambda }{\lambda }},1/2\,{\frac {2\,a-\lambda }{\lambda }},\sin \left ( \lambda \,x \right ) \right ) \cos \left ( \lambda \,x \right ) y \right ) ^{-1}} \right ) \]
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Added January 14, 2019.
Problem 2.6.3.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x +\left ( y^2+ a x \tan ^k(b x) y + a \tan ^k(b x) \right ) w_y = 0 \]
Mathematica ✗
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B]; pde = D[w[x, y], x] + (y^2 + a*x*Tan[b*x]^k*y + a*Tan[b*x]^k)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \text {Failed} \] Timed out
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';B:='B';mu:='mu';d:='d';s:='s'; pde := diff(w(x,y),x)+( y^2+ a*x *tan(b*x)^k * y + a*tan(b*x)^k)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1}{yx+1} \left ( yx\int \!{{\rm e}^{\int \!{\frac {a \left ( \tan \left ( bx \right ) \right ) ^{k}{x}^{2}-2}{x}}\,{\rm d}x}}\,{\rm d}x+{{\rm e}^{\int \!{\frac {a \left ( \tan \left ( bx \right ) \right ) ^{k}{x}^{2}-2}{x}}\,{\rm d}x}}x+\int \!{{\rm e}^{\int \!{\frac {a \left ( \tan \left ( bx \right ) \right ) ^{k}{x}^{2}-2}{x}}\,{\rm d}x}}\,{\rm d}x \right ) } \right ) \]
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Added January 14, 2019.
Problem 2.6.3.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x -\left ( (k+1) x^k y^2- a x^{k+1} (\tan x)^m y + a(\tan x)^m \right ) w_y = 0 \]
Mathematica ✗
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B]; pde = D[w[x, y], x] - ((k + 1)*x^k*y^2 - a*x^(k + 1)*Tan[x]^m*y + a*Tan[x]^m)*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'; C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s'; pde := diff(w(x,y),x)-( (k+1)*x^k*y^2- a*x^(k+1)*tan(x)^m*y + a*tan(x)^m )*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ \text {server hangs} \] Server hangs
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Added January 20, 2019.
Problem 2.6.3.8 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x +\left ( a \tan ^n(\lambda x) y^2- a b^2 \tan ^{n+2}(\lambda x) + b \lambda \tan ^2(\lambda x)+ b \lambda \right ) w_y = 0 \]
Mathematica ✗
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B]; pde = D[w[x, y], x] + (a*Tan[lambda*x]^n*y^2 - a*b^2*Tan[lambda*x]^(n + 2) + b*lambda*Tan[lambda*x]^2 + b*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'; C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s'; pde := diff(w(x,y),x)+(a*tan(lambda*x)^n*y^2- a*b^2*tan(lambda*x)^(n+2) + b*lambda*tan(lambda*x)^2+ b*lambda)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ \text {server hangs} \] Server hangs
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Added January 20, 2019.
Problem 2.6.3.9 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x +\left ( a \tan ^k(\lambda x+\mu )(y-b x^n-c)^2 + y- b x^n + b n x^{n-1}-c \right ) w_y = 0 \]
Mathematica ✗
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B]; pde = D[w[x, y], x] + (a*Tan[lambda*x + mu]^k*(y - b*x^n - c)^2 + y - b*x^n + b*n*x^(n - 1) - c)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \text {\$Aborted} \] Timed out
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';B:='B';mu:='mu';d:='d';s:='s'; pde := diff(w(x,y),x)+(a *tan(lambda*x+mu)^k*(y-b*x^n-c)^2 + y- b*x^n + b*n*x^(n-1)-c)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ \text { Exception } \] Timed out
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Added January 20, 2019.
Problem 2.6.3.10 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ x w_x +\left ( a \tan ^m(\lambda x)y^2 +k y+ a b^2 x^{2 k} \tan ^m(\lambda x) \right ) w_y = 0 \]
Mathematica ✓
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B]; pde = x*D[w[x, y], x] + (a*Tan[lambda*x]^m*y^2 + k*y + a*b^2*x^(2*k)*Tan[lambda*x]^m)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to c_1\left (\tan ^{-1}\left (\frac {y x^{-k}}{\sqrt {b^2}}\right )-\sqrt {b^2} \int _1^x a K[1]^{k-1} \tan ^m(\lambda K[1]) \, dK[1]\right )\right \}\right \} \]
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'; C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s'; pde := x*diff(w(x,y),x)+ (a*tan(lambda*x)^m*y^2 +k*y+ a*b^2*x^(2*k)*tan(lambda*x)^m )*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( ab\int \!{x}^{k-1} \left ( \tan \left ( \lambda \,x \right ) \right ) ^{m}\,{\rm d}x-\arctan \left ( {\frac {{x}^{-k}y}{b}} \right ) \right ) \]
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Added January 20, 2019.
Problem 2.6.3.11 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ (a \tan (\lambda x)+b) w_x +\left ( y^2+ c \tan (\mu x) y - k^2 + c k \tan (\mu x) \right ) w_y = 0 \]
Mathematica ✗
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B]; pde = (a*Tan[lambda*x] + b)*D[w[x, y], x] + (y^2 + c*Tan[mu*x]*y - k^2 + c*k*Tan[mu*x])*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \text {\$Aborted} \] Timed out
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';B:='B';mu:='mu';d:='d';s:='s'; pde := (a*tan(lambda*x)+b)*diff(w(x,y),x)+ (y^2+ c *tan(mu*x)*y - k^2 + c*k*tan(mu*x) )*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime')); sol:=simplify(sol);
\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1}{k+y} \left ( \left ( k+y \right ) \int \!{\frac {\cos \left ( \lambda \,x \right ) }{\sin \left ( \lambda \,x \right ) a+b\cos \left ( \lambda \,x \right ) } \left ( {{\rm e}^{2\,i\mu \,x}}+1 \right ) ^{{\frac {-ic}{ \left ( a+ib \right ) \mu }}} \left ( {\frac {\sin \left ( \lambda \,x \right ) a+b\cos \left ( \lambda \,x \right ) }{\cos \left ( \lambda \,x \right ) }} \right ) ^{-2\,{\frac {ak}{\lambda \, \left ( {a}^{2}+{b}^{2} \right ) }}} \left ( \left ( \cos \left ( \lambda \,x \right ) \right ) ^{-2} \right ) ^{{\frac {ak}{\lambda \, \left ( {a}^{2}+{b}^{2} \right ) }}}{{\rm e}^{{\frac {1}{ \left ( {a}^{2}+{b}^{2} \right ) \left ( a+ib \right ) \lambda } \left ( 2\, \left ( a+ib \right ) ac \left ( {a}^{2}+{b}^{2} \right ) \lambda \,\int \!{\frac {{{\rm e}^{2\,i\mu \,x}}-1}{ \left ( a+ib \right ) \left ( {{\rm e}^{2\,i\mu \,x}}+1 \right ) \left ( \left ( a+ib \right ) {{\rm e}^{2\,i\lambda \,x}}+ib-a \right ) }}\,{\rm d}x-2\,k \left ( a+ib \right ) b\arctan \left ( {\frac {\sin \left ( \lambda \,x \right ) }{\cos \left ( \lambda \,x \right ) }} \right ) -cx\lambda \, \left ( {a}^{2}+{b}^{2} \right ) \right ) }}}}\,{\rm d}x+ \left ( {{\rm e}^{2\,i\mu \,x}}+1 \right ) ^{{\frac {-ic}{ \left ( a+ib \right ) \mu }}} \left ( {\frac {\sin \left ( \lambda \,x \right ) a+b\cos \left ( \lambda \,x \right ) }{\cos \left ( \lambda \,x \right ) }} \right ) ^{-2\,{\frac {ak}{\lambda \, \left ( {a}^{2}+{b}^{2} \right ) }}} \left ( \left ( \cos \left ( \lambda \,x \right ) \right ) ^{-2} \right ) ^{{\frac {ak}{\lambda \, \left ( {a}^{2}+{b}^{2} \right ) }}}{{\rm e}^{{\frac {1}{ \left ( {a}^{2}+{b}^{2} \right ) \left ( a+ib \right ) \lambda } \left ( 2\, \left ( a+ib \right ) ac \left ( {a}^{2}+{b}^{2} \right ) \lambda \,\int \!{\frac {{{\rm e}^{2\,i\mu \,x}}-1}{ \left ( a+ib \right ) \left ( {{\rm e}^{2\,i\mu \,x}}+1 \right ) \left ( \left ( a+ib \right ) {{\rm e}^{2\,i\lambda \,x}}+ib-a \right ) }}\,{\rm d}x-2\,k \left ( a+ib \right ) b\arctan \left ( {\frac {\sin \left ( \lambda \,x \right ) }{\cos \left ( \lambda \,x \right ) }} \right ) -cx\lambda \, \left ( {a}^{2}+{b}^{2} \right ) \right ) }}} \right ) } \right ) \]
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Added January 20, 2019.
Problem 2.6.3.12 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ (a x^n y^m + b x) w_x + \tan ^k(\lambda y) w_y = 0 \]
Mathematica ✗
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B]; pde = (a*x^n*y^m + b*x)*D[w[x, y], x] + Tan[lambda*y]^k*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'; C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s'; pde := (a*x^n*y^m + b*x)*diff(w(x,y),x)+ tan(lambda*y)^k*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( {x}^{-n+1}{{\rm e}^{b\int \! \left ( \tan \left ( y\lambda \right ) \right ) ^{-k}\,{\rm d}y \left ( n-1 \right ) }}+an\int \!{{\rm e}^{b\int \! \left ( \tan \left ( y\lambda \right ) \right ) ^{-k}\,{\rm d}y \left ( n-1 \right ) }}{y}^{m} \left ( \tan \left ( y\lambda \right ) \right ) ^{-k}\,{\rm d}y-a\int \!{{\rm e}^{b\int \! \left ( \tan \left ( y\lambda \right ) \right ) ^{-k}\,{\rm d}y \left ( n-1 \right ) }}{y}^{m} \left ( \tan \left ( y\lambda \right ) \right ) ^{-k}\,{\rm d}y \right ) \]
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Added January 20, 2019.
Problem 2.6.3.13 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ (a x^n + b x \tan ^m y) w_x + y^k w_y = 0 \]
Mathematica ✗
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B]; pde = (a*x^n + b*x*Tan[y]^m)*D[w[x, y], x] + y^k*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \text {\$Aborted} \] Timed out
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';B:='B';mu:='mu';d:='d';s:='s'; pde := (a*x^n + b*x*tan(y)^m)*diff(w(x,y),x)+ y^k*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( {x}^{-n+1}{{\rm e}^{b\int \! \left ( \tan \left ( y \right ) \right ) ^{m}{y}^{-k}\,{\rm d}y \left ( n-1 \right ) }}+an\int \!{{\rm e}^{b\int \! \left ( \tan \left ( y \right ) \right ) ^{m}{y}^{-k}\,{\rm d}y \left ( n-1 \right ) }}{y}^{-k}\,{\rm d}y-a\int \!{{\rm e}^{b\int \! \left ( \tan \left ( y \right ) \right ) ^{m}{y}^{-k}\,{\rm d}y \left ( n-1 \right ) }}{y}^{-k}\,{\rm d}y \right ) \]
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Added January 20, 2019.
Problem 2.6.3.14 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ (a x^n + b x \tan ^m y) w_x + \tan ^k(\lambda y) w_y = 0 \]
Mathematica ✗
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B]; pde = (a*x^n + b*x*Tan[y]^m)*D[w[x, y], x] + Tan[lambda*y]^k*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \text {\$Aborted} \] Timed out
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';B:='B';mu:='mu';d:='d';s:='s'; pde := (a*x^n + b*x*tan(y)^m)*diff(w(x,y),x)+ tan(lambda*y)^k*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( {x}^{-n+1}{{\rm e}^{b\int \! \left ( \tan \left ( y \right ) \right ) ^{m} \left ( \tan \left ( y\lambda \right ) \right ) ^{-k}\,{\rm d}y \left ( n-1 \right ) }}+an\int \!{{\rm e}^{b\int \! \left ( \tan \left ( y \right ) \right ) ^{m} \left ( \tan \left ( y\lambda \right ) \right ) ^{-k}\,{\rm d}y \left ( n-1 \right ) }} \left ( \tan \left ( y\lambda \right ) \right ) ^{-k}\,{\rm d}y-a\int \!{{\rm e}^{b\int \! \left ( \tan \left ( y \right ) \right ) ^{m} \left ( \tan \left ( y\lambda \right ) \right ) ^{-k}\,{\rm d}y \left ( n-1 \right ) }} \left ( \tan \left ( y\lambda \right ) \right ) ^{-k}\,{\rm d}y \right ) \]
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Added January 20, 2019.
Problem 2.6.3.15 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ (a x^n \tan ^m y + b x) w_x + \tan ^k(\lambda y) w_y = 0 \]
Mathematica ✗
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B]; pde = (a*x^n*Tan[y]^m + b*x)*D[w[x, y], x] + Tan[lambda*y]^k*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \text {Failed} \] Timed out
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';B:='B';mu:='mu';d:='d';s:='s'; pde := (a*x^n*tan(y)^m+ b*x)*diff(w(x,y),x)+ tan(lambda*y)^k*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( {x}^{-n+1}{{\rm e}^{b\int \! \left ( \tan \left ( y\lambda \right ) \right ) ^{-k}\,{\rm d}y \left ( n-1 \right ) }}+an\int \!{{\rm e}^{b\int \! \left ( \tan \left ( y\lambda \right ) \right ) ^{-k}\,{\rm d}y \left ( n-1 \right ) }} \left ( \tan \left ( y \right ) \right ) ^{m} \left ( \tan \left ( y\lambda \right ) \right ) ^{-k}\,{\rm d}y-a\int \!{{\rm e}^{b\int \! \left ( \tan \left ( y\lambda \right ) \right ) ^{-k}\,{\rm d}y \left ( n-1 \right ) }} \left ( \tan \left ( y \right ) \right ) ^{m} \left ( \tan \left ( y\lambda \right ) \right ) ^{-k}\,{\rm d}y \right ) \]