____________________________________________________________________________________
Added January 14, 2019.
Problem 2.6.2.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x +\left ( a \cos ^k(\lambda x)+b \right ) w_y = 0 \]
Mathematica ✓
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d]; pde = D[w[x, y], x] + (a*Cos[lambda*x]^k + b)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to c_1\left (\frac {a \sin (\lambda x) \cos ^{k+1}(\lambda x) \text {Hypergeometric2F1}\left (\frac {1}{2},\frac {k+1}{2},\frac {k+3}{2},\cos ^2(\lambda x)\right )}{(k+1) \lambda \sqrt {\sin ^2(\lambda x)}}-b x+y\right )\right \}\right \} \]
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*cos(lambda*x)^k+b)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( -bx-\int \!a \left ( \cos \left ( \lambda \,x \right ) \right ) ^{k}\,{\rm d}x+y \right ) \] Contains unresolved integral
____________________________________________________________________________________
Added January 14, 2019.
Problem 2.6.2.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x +\left ( a \cos ^k(\lambda y)+b \right ) w_y = 0 \]
Mathematica ✓
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d]; pde = D[w[x, y], x] + (a*Cos[lambda*y]^k + b)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to c_1\left (\int _1^y \frac {1}{a \cos ^k(\lambda K[1])+b} \, dK[1]-x\right )\right \}\right \} \]
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*cos(lambda*y)^k+b)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( -\int \! \left ( a \left ( \cos \left ( y\lambda \right ) \right ) ^{k}+b \right ) ^{-1}\,{\rm d}y+x \right ) \]
____________________________________________________________________________________
Added January 14, 2019.
Problem 2.6.2.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x +a \cos ^k(\lambda x) \cos ^n(\mu y) w_y = 0 \]
Mathematica ✓
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d]; pde = D[w[x, y], x] + a*Cos[lambda*y]^k*Cos[mu*y]^n*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to c_1\left (\int _1^y \cos ^{-k}(\lambda K[1]) \cos ^{-n}(\mu K[1]) \, dK[1]-a x\right )\right \}\right \} \]
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*cos(lambda*y)^k*cos(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'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {ax-\int \! \left ( \cos \left ( y\lambda \right ) \right ) ^{-k} \left ( \cos \left ( \mu \,y \right ) \right ) ^{-n}\,{\rm d}y}{a}} \right ) \]
____________________________________________________________________________________
Added January 14, 2019.
Problem 2.6.2.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x +a \cos ^k(x+\lambda y) w_y = 0 \]
Mathematica ✗
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d]; pde = D[w[x, y], x] + a*Cos[x + 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 := diff(w(x,y),x)+a*cos(x+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'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( -\int ^{{\frac {y\lambda +x}{\lambda }}}\! \left ( 1+a \left ( \cos \left ( \lambda \,{\it \_a} \right ) \right ) ^{k}\lambda \right ) ^{-1}{d{\it \_a}}\lambda +x \right ) \]
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Added January 14, 2019.
Problem 2.6.2.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x +\left ( y^2-a^2 + a \lambda \cos (\lambda x) + a^2 \cos ^2(\lambda x) \right ) w_y = 0 \]
Mathematica ✗
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d]; pde = D[w[x, y], x] + (y^2 - a^2 + a*lambda*Cos[lambda*x] + a^2*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'; C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s'; pde := diff(w(x,y),x)+( y^2-a^2 + a *lambda*cos(lambda*x) + a^2*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'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( -1/2\,{\sqrt {2\,\cos \left ( \lambda \,x \right ) +2} \left ( 2\,\sin \left ( \lambda \,x \right ) \HeunC \left ( 4\,{\frac {a}{\lambda }},-1/2,-1/2,-2\,{\frac {a}{\lambda }},1/8\,{\frac {8\,a+3\,\lambda }{\lambda }},1/2\,\cos \left ( \lambda \,x \right ) +1/2 \right ) a+\sin \left ( \lambda \,x \right ) \lambda \,\HeunCPrime \left ( 4\,{\frac {a}{\lambda }},-1/2,-1/2,-2\,{\frac {a}{\lambda }},1/8\,{\frac {8\,a+3\,\lambda }{\lambda }},1/2\,\cos \left ( \lambda \,x \right ) +1/2 \right ) -2\,y\HeunC \left ( 4\,{\frac {a}{\lambda }},-1/2,-1/2,-2\,{\frac {a}{\lambda }},1/8\,{\frac {8\,a+3\,\lambda }{\lambda }},1/2\,\cos \left ( \lambda \,x \right ) +1/2 \right ) \right ) \left ( 2\,\sin \left ( \lambda \,x \right ) \cos \left ( \lambda \,x \right ) \HeunC \left ( 4\,{\frac {a}{\lambda }},1/2,-1/2,-2\,{\frac {a}{\lambda }},1/8\,{\frac {8\,a+3\,\lambda }{\lambda }},1/2\,\cos \left ( \lambda \,x \right ) +1/2 \right ) a+\sin \left ( \lambda \,x \right ) \cos \left ( \lambda \,x \right ) \HeunCPrime \left ( 4\,{\frac {a}{\lambda }},1/2,-1/2,-2\,{\frac {a}{\lambda }},1/8\,{\frac {8\,a+3\,\lambda }{\lambda }},1/2\,\cos \left ( \lambda \,x \right ) +1/2 \right ) \lambda +2\,\sin \left ( \lambda \,x \right ) \HeunC \left ( 4\,{\frac {a}{\lambda }},1/2,-1/2,-2\,{\frac {a}{\lambda }},1/8\,{\frac {8\,a+3\,\lambda }{\lambda }},1/2\,\cos \left ( \lambda \,x \right ) +1/2 \right ) a+\sin \left ( \lambda \,x \right ) \HeunC \left ( 4\,{\frac {a}{\lambda }},1/2,-1/2,-2\,{\frac {a}{\lambda }},1/8\,{\frac {8\,a+3\,\lambda }{\lambda }},1/2\,\cos \left ( \lambda \,x \right ) +1/2 \right ) \lambda +\sin \left ( \lambda \,x \right ) \lambda \,\HeunCPrime \left ( 4\,{\frac {a}{\lambda }},1/2,-1/2,-2\,{\frac {a}{\lambda }},1/8\,{\frac {8\,a+3\,\lambda }{\lambda }},1/2\,\cos \left ( \lambda \,x \right ) +1/2 \right ) -2\,y\HeunC \left ( 4\,{\frac {a}{\lambda }},1/2,-1/2,-2\,{\frac {a}{\lambda }},1/8\,{\frac {8\,a+3\,\lambda }{\lambda }},1/2\,\cos \left ( \lambda \,x \right ) +1/2 \right ) \cos \left ( \lambda \,x \right ) -2\,y\HeunC \left ( 4\,{\frac {a}{\lambda }},1/2,-1/2,-2\,{\frac {a}{\lambda }},1/8\,{\frac {8\,a+3\,\lambda }{\lambda }},1/2\,\cos \left ( \lambda \,x \right ) +1/2 \right ) \right ) ^{-1}} \right ) \]
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Added January 14, 2019.
Problem 2.6.2.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x +\left ( \lambda \cos (\lambda x) y^2 + \lambda \cos ^3(\lambda x) \right ) w_y = 0 \]
Mathematica ✗
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d]; pde = D[w[x, y], x] + (lambda*Cos[lambda*x]*y^2 + lambda*Cos[lambda*x]^3)*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)+(lambda*cos(lambda*x)*y^2 + lambda*cos(lambda*x)^3)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac {{{\sl M}\left (1,\,3/2,\,- \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}\right )} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}-\sin \left ( \lambda \,x \right ) {{\sl M}\left (1,\,3/2,\,- \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}\right )}y-1}{2+ \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}{{\sl U}\left (1,\,3/2,\,- \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}\right )}-\sin \left ( \lambda \,x \right ) {{\sl U}\left (1,\,3/2,\,- \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}\right )}y}} \right ) \]
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Added January 14, 2019.
Problem 2.6.2.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ 2 w_x +\left ( (\lambda +a+a \cos (\lambda x)) y^2 + \lambda - a + a \cos (\lambda x) \right ) w_y = 0 \]
Mathematica ✗
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d]; pde = 2*D[w[x, y], x] + ((lambda + a + a*Cos[lambda*x])*y^2 + lambda - a + a*Cos[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'; C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s'; pde := 2*diff(w(x,y),x)+ ((lambda+a+a*cos(lambda*x))*y^2 +lambda - a + a *cos(lambda*x))*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac { \left ( -y\cos \left ( \lambda \,x \right ) +\sin \left ( \lambda \,x \right ) -y \right ) \sqrt {\cos \left ( \lambda \,x \right ) +1}\sqrt {\cos \left ( \lambda \,x \right ) -1}}{\lambda }{{\rm e}^{{\frac {\cos \left ( \lambda \,x \right ) a}{\lambda }}}} \left ( -\sqrt {\cos \left ( \lambda \,x \right ) -1}\sqrt {\cos \left ( \lambda \,x \right ) +1}y{{\rm e}^{{\frac {\cos \left ( \lambda \,x \right ) a}{\lambda }}}}\int \!{\frac { \left ( \lambda +a+\cos \left ( \lambda \,x \right ) a \right ) \sin \left ( \lambda \,x \right ) }{\sqrt {\cos \left ( \lambda \,x \right ) -1} \left ( \cos \left ( \lambda \,x \right ) +1 \right ) ^{3/2}}{{\rm e}^{-{\frac {\cos \left ( \lambda \,x \right ) a}{\lambda }}}}}\,{\rm d}x\cos \left ( \lambda \,x \right ) +\sin \left ( \lambda \,x \right ) \sqrt {\cos \left ( \lambda \,x \right ) -1}\sqrt {\cos \left ( \lambda \,x \right ) +1}{{\rm e}^{{\frac {\cos \left ( \lambda \,x \right ) a}{\lambda }}}}\int \!{\frac { \left ( \lambda +a+\cos \left ( \lambda \,x \right ) a \right ) \sin \left ( \lambda \,x \right ) }{\sqrt {\cos \left ( \lambda \,x \right ) -1} \left ( \cos \left ( \lambda \,x \right ) +1 \right ) ^{3/2}}{{\rm e}^{-{\frac {\cos \left ( \lambda \,x \right ) a}{\lambda }}}}}\,{\rm d}x-\sqrt {\cos \left ( \lambda \,x \right ) -1}\sqrt {\cos \left ( \lambda \,x \right ) +1}y{{\rm e}^{{\frac {\cos \left ( \lambda \,x \right ) a}{\lambda }}}}\int \!{\frac { \left ( \lambda +a+\cos \left ( \lambda \,x \right ) a \right ) \sin \left ( \lambda \,x \right ) }{\sqrt {\cos \left ( \lambda \,x \right ) -1} \left ( \cos \left ( \lambda \,x \right ) +1 \right ) ^{3/2}}{{\rm e}^{-{\frac {\cos \left ( \lambda \,x \right ) a}{\lambda }}}}}\,{\rm d}x-2\,\sin \left ( \lambda \,x \right ) \right ) ^{-1}} \right ) \]
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Added January 14, 2019.
Problem 2.6.2.8 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x +\left ( (\lambda +a \cos ^2(\lambda x)) y^2 + \lambda - a + a \cos ^2(\lambda x) \right ) w_y = 0 \]
Mathematica ✗
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d]; pde = D[w[x, y], x] + ((lambda + a*Cos[lambda*x]^2)*y^2 + lambda - a + a*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'; C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s'; pde := diff(w(x,y),x)+ ((lambda+a*cos(lambda*x)^2)*y^2 + lambda - a + a*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'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( 1/2\,{\frac {\sqrt {-1+\cos \left ( 2\,\lambda \,x \right ) } \left ( -8\, \left ( \cos \left ( \lambda \,x \right ) \right ) ^{6}ya-8\, \left ( \cos \left ( \lambda \,x \right ) \right ) ^{4}y\lambda +\sin \left ( 2\,\lambda \,x \right ) \left ( \cos \left ( 2\,\lambda \,x \right ) \right ) ^{2}a+2\,\sin \left ( 2\,\lambda \,x \right ) \cos \left ( 2\,\lambda \,x \right ) a+2\,\sin \left ( 2\,\lambda \,x \right ) \cos \left ( 2\,\lambda \,x \right ) \lambda +a\sin \left ( 2\,\lambda \,x \right ) +2\,\lambda \,\sin \left ( 2\,\lambda \,x \right ) \right ) }{\lambda }{{\rm e}^{1/2\,{\frac {\cos \left ( 2\,\lambda \,x \right ) a}{\lambda }}}} \left ( -8\,\sqrt {-1+\cos \left ( 2\,\lambda \,x \right ) }\int \!{\frac { \left ( \cos \left ( 2\,\lambda \,x \right ) a+a+2\,\lambda \right ) \sin \left ( 2\,\lambda \,x \right ) }{ \left ( \cos \left ( 2\,\lambda \,x \right ) +1 \right ) ^{3/2}\sqrt {-1+\cos \left ( 2\,\lambda \,x \right ) }}{{\rm e}^{-1/2\,{\frac {\cos \left ( 2\,\lambda \,x \right ) a}{\lambda }}}}}\,{\rm d}x{{\rm e}^{1/2\,{\frac {\cos \left ( 2\,\lambda \,x \right ) a}{\lambda }}}} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{6}ya-8\,\sqrt {-1+\cos \left ( 2\,\lambda \,x \right ) }\int \!{\frac { \left ( \cos \left ( 2\,\lambda \,x \right ) a+a+2\,\lambda \right ) \sin \left ( 2\,\lambda \,x \right ) }{ \left ( \cos \left ( 2\,\lambda \,x \right ) +1 \right ) ^{3/2}\sqrt {-1+\cos \left ( 2\,\lambda \,x \right ) }}{{\rm e}^{-1/2\,{\frac {\cos \left ( 2\,\lambda \,x \right ) a}{\lambda }}}}}\,{\rm d}x{{\rm e}^{1/2\,{\frac {\cos \left ( 2\,\lambda \,x \right ) a}{\lambda }}}} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{4}y\lambda +\sqrt {-1+\cos \left ( 2\,\lambda \,x \right ) }\int \!{\frac { \left ( \cos \left ( 2\,\lambda \,x \right ) a+a+2\,\lambda \right ) \sin \left ( 2\,\lambda \,x \right ) }{ \left ( \cos \left ( 2\,\lambda \,x \right ) +1 \right ) ^{3/2}\sqrt {-1+\cos \left ( 2\,\lambda \,x \right ) }}{{\rm e}^{-1/2\,{\frac {\cos \left ( 2\,\lambda \,x \right ) a}{\lambda }}}}}\,{\rm d}x\sin \left ( 2\,\lambda \,x \right ) {{\rm e}^{1/2\,{\frac {\cos \left ( 2\,\lambda \,x \right ) a}{\lambda }}}} \left ( \cos \left ( 2\,\lambda \,x \right ) \right ) ^{2}a+2\,\sqrt {-1+\cos \left ( 2\,\lambda \,x \right ) }\int \!{\frac { \left ( \cos \left ( 2\,\lambda \,x \right ) a+a+2\,\lambda \right ) \sin \left ( 2\,\lambda \,x \right ) }{ \left ( \cos \left ( 2\,\lambda \,x \right ) +1 \right ) ^{3/2}\sqrt {-1+\cos \left ( 2\,\lambda \,x \right ) }}{{\rm e}^{-1/2\,{\frac {\cos \left ( 2\,\lambda \,x \right ) a}{\lambda }}}}}\,{\rm d}x\sin \left ( 2\,\lambda \,x \right ) {{\rm e}^{1/2\,{\frac {\cos \left ( 2\,\lambda \,x \right ) a}{\lambda }}}}\cos \left ( 2\,\lambda \,x \right ) a+2\,\sqrt {-1+\cos \left ( 2\,\lambda \,x \right ) }\int \!{\frac { \left ( \cos \left ( 2\,\lambda \,x \right ) a+a+2\,\lambda \right ) \sin \left ( 2\,\lambda \,x \right ) }{ \left ( \cos \left ( 2\,\lambda \,x \right ) +1 \right ) ^{3/2}\sqrt {-1+\cos \left ( 2\,\lambda \,x \right ) }}{{\rm e}^{-1/2\,{\frac {\cos \left ( 2\,\lambda \,x \right ) a}{\lambda }}}}}\,{\rm d}x\sin \left ( 2\,\lambda \,x \right ) {{\rm e}^{1/2\,{\frac {\cos \left ( 2\,\lambda \,x \right ) a}{\lambda }}}}\cos \left ( 2\,\lambda \,x \right ) \lambda +\sqrt {-1+\cos \left ( 2\,\lambda \,x \right ) }\int \!{\frac { \left ( \cos \left ( 2\,\lambda \,x \right ) a+a+2\,\lambda \right ) \sin \left ( 2\,\lambda \,x \right ) }{ \left ( \cos \left ( 2\,\lambda \,x \right ) +1 \right ) ^{3/2}\sqrt {-1+\cos \left ( 2\,\lambda \,x \right ) }}{{\rm e}^{-1/2\,{\frac {\cos \left ( 2\,\lambda \,x \right ) a}{\lambda }}}}}\,{\rm d}x\sin \left ( 2\,\lambda \,x \right ) {{\rm e}^{1/2\,{\frac {\cos \left ( 2\,\lambda \,x \right ) a}{\lambda }}}}a+2\,\sqrt {-1+\cos \left ( 2\,\lambda \,x \right ) }\int \!{\frac { \left ( \cos \left ( 2\,\lambda \,x \right ) a+a+2\,\lambda \right ) \sin \left ( 2\,\lambda \,x \right ) }{ \left ( \cos \left ( 2\,\lambda \,x \right ) +1 \right ) ^{3/2}\sqrt {-1+\cos \left ( 2\,\lambda \,x \right ) }}{{\rm e}^{-1/2\,{\frac {\cos \left ( 2\,\lambda \,x \right ) a}{\lambda }}}}}\,{\rm d}x\sin \left ( 2\,\lambda \,x \right ) {{\rm e}^{1/2\,{\frac {\cos \left ( 2\,\lambda \,x \right ) a}{\lambda }}}}\lambda -2\,\sqrt {\cos \left ( 2\,\lambda \,x \right ) +1}\sin \left ( 2\,\lambda \,x \right ) \cos \left ( 2\,\lambda \,x \right ) a-2\,\sqrt {\cos \left ( 2\,\lambda \,x \right ) +1}\sin \left ( 2\,\lambda \,x \right ) a-4\,\sqrt {\cos \left ( 2\,\lambda \,x \right ) +1}\sin \left ( 2\,\lambda \,x \right ) \lambda \right ) ^{-1}} \right ) \]
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Added January 14, 2019.
Problem 2.6.2.9 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ (a x^n y^m+b x) w_x +\cos ^k(\lambda y) w_y = 0 \]
Mathematica ✗
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d]; pde = (a*x^n*y^m + b*x)*D[w[x, y], x] + Cos[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*y^m+b*x)*diff(w(x,y),x)+ cos(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'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( {x}^{-n+1}{{\rm e}^{b\int \! \left ( \cos \left ( y\lambda \right ) \right ) ^{-k}\,{\rm d}y \left ( n-1 \right ) }}+an\int \!{{\rm e}^{b\int \! \left ( \cos \left ( y\lambda \right ) \right ) ^{-k}\,{\rm d}y \left ( n-1 \right ) }}{y}^{m} \left ( \cos \left ( y\lambda \right ) \right ) ^{-k}\,{\rm d}y-a\int \!{{\rm e}^{b\int \! \left ( \cos \left ( y\lambda \right ) \right ) ^{-k}\,{\rm d}y \left ( n-1 \right ) }}{y}^{m} \left ( \cos \left ( y\lambda \right ) \right ) ^{-k}\,{\rm d}y \right ) \]
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Added January 14, 2019.
Problem 2.6.2.10 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ (a x^n +b x \cos ^m y) w_x +y^k w_y = 0 \]
Mathematica ✗
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d]; pde = (a*x^n + b*x*Cos[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*cos(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'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( {x}^{-n+1}{{\rm e}^{b\int \! \left ( \cos \left ( y \right ) \right ) ^{m}{y}^{-k}\,{\rm d}y \left ( n-1 \right ) }}+an\int \!{{\rm e}^{b\int \! \left ( \cos \left ( y \right ) \right ) ^{m}{y}^{-k}\,{\rm d}y \left ( n-1 \right ) }}{y}^{-k}\,{\rm d}y-a\int \!{{\rm e}^{b\int \! \left ( \cos \left ( y \right ) \right ) ^{m}{y}^{-k}\,{\rm d}y \left ( n-1 \right ) }}{y}^{-k}\,{\rm d}y \right ) \]
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Added January 14, 2019.
Problem 2.6.2.11 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ (a x^n +b x \cos ^m y) w_x + \cos ^k(\lambda y) w_y = 0 \]
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*Cos[y]^m)*D[w[x, y], x] + Cos[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*cos(y)^m)*diff(w(x,y),x)+cos(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'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( {x}^{-n+1}{{\rm e}^{b\int \! \left ( \cos \left ( y \right ) \right ) ^{m} \left ( \cos \left ( y\lambda \right ) \right ) ^{-k}\,{\rm d}y \left ( n-1 \right ) }}+an\int \!{{\rm e}^{b\int \! \left ( \cos \left ( y \right ) \right ) ^{m} \left ( \cos \left ( y\lambda \right ) \right ) ^{-k}\,{\rm d}y \left ( n-1 \right ) }} \left ( \cos \left ( y\lambda \right ) \right ) ^{-k}\,{\rm d}y-a\int \!{{\rm e}^{b\int \! \left ( \cos \left ( y \right ) \right ) ^{m} \left ( \cos \left ( y\lambda \right ) \right ) ^{-k}\,{\rm d}y \left ( n-1 \right ) }} \left ( \cos \left ( y\lambda \right ) \right ) ^{-k}\,{\rm d}y \right ) \]
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Added January 14, 2019.
Problem 2.6.2.12 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ (a x^n \cos ^m y + b x) w_x + \cos ^k(\lambda y) w_y = 0 \]
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*Cos[y]^m + b*x)*D[w[x, y], x] + Cos[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*cos(y)^m+b*x)*diff(w(x,y),x)+cos(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'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( {x}^{-n+1}{{\rm e}^{b\int \! \left ( \cos \left ( y\lambda \right ) \right ) ^{-k}\,{\rm d}y \left ( n-1 \right ) }}+an\int \!{{\rm e}^{b\int \! \left ( \cos \left ( y\lambda \right ) \right ) ^{-k}\,{\rm d}y \left ( n-1 \right ) }} \left ( \cos \left ( y \right ) \right ) ^{m} \left ( \cos \left ( y\lambda \right ) \right ) ^{-k}\,{\rm d}y-a\int \!{{\rm e}^{b\int \! \left ( \cos \left ( y\lambda \right ) \right ) ^{-k}\,{\rm d}y \left ( n-1 \right ) }} \left ( \cos \left ( y \right ) \right ) ^{m} \left ( \cos \left ( y\lambda \right ) \right ) ^{-k}\,{\rm d}y \right ) \]