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Added January 20, 2019.
Problem 2.6.5.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + a \sin ^k(\lambda x) \cos ^n(\mu y) w_y = 0 \]
Mathematica ✓
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B, v]; pde = D[w[x, y], x] + a*Sin[lambda*x]^k*Cos[mu*y]^n*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to c_1\left (\frac {a \cos (\lambda x) \sin ^{k+1}(\lambda x) \sin ^2(\lambda x)^{-\frac {k}{2}-\frac {1}{2}} \text {Hypergeometric2F1}\left (\frac {1}{2},\frac {1-k}{2},\frac {3}{2},\cos ^2(\lambda x)\right )+\frac {\lambda \sqrt {\sin ^2(\mu y)} \csc (\mu y) \cos ^{1-n}(\mu y) \text {Hypergeometric2F1}\left (\frac {1}{2},\frac {1-n}{2},\frac {3-n}{2},\cos ^2(\mu y)\right )}{\mu (n-1)}}{\lambda }\right )\right \}\right \} \]
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';v:='v'; pde := diff(w(x,y),x)+ a*sin(lambda*x)^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'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( -\int \! \left ( \sin \left ( \lambda \,x \right ) \right ) ^{k}\,{\rm d}x+\int \!{\frac { \left ( \cos \left ( \mu \,y \right ) \right ) ^{-n}}{a}}\,{\rm d}y \right ) \] Has unresolved integrals
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Added January 20, 2019.
Problem 2.6.5.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + \left ( y^2-y \tan x+a(1-a) \cot ^2 x \right ) w_y = 0 \]
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]; pde = D[w[x, y], x] + (y^2 - y*Tan[x] + a*(1 - a)*Cot[x]^2)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to c_1\left (\frac {2 y \cos ^2(x) \left (\cos ^2(x)-1\right )^{\frac {1}{4} i \left (\sqrt {-\frac {1}{(a-1) a}-4}-\frac {i}{\sqrt {a-1} \sqrt {a}}\right ) \sqrt {a-1} \sqrt {a}}-2 y \left (\cos ^2(x)-1\right )^{\frac {1}{4} i \left (\sqrt {-\frac {1}{(a-1) a}-4}-\frac {i}{\sqrt {a-1} \sqrt {a}}\right ) \sqrt {a-1} \sqrt {a}}-i \sqrt {-\frac {1}{(a-1) a}-4} \sqrt {a-1} \sqrt {a} \sin (x) \cos (x) \left (\cos ^2(x)-1\right )^{\frac {1}{4} i \left (\sqrt {-\frac {1}{(a-1) a}-4}-\frac {i}{\sqrt {a-1} \sqrt {a}}\right ) \sqrt {a-1} \sqrt {a}}-\sin (x) \cos (x) \left (\cos ^2(x)-1\right )^{\frac {1}{4} i \left (\sqrt {-\frac {1}{(a-1) a}-4}-\frac {i}{\sqrt {a-1} \sqrt {a}}\right ) \sqrt {a-1} \sqrt {a}}}{-2 y \cos ^2(x) \left (\cos ^2(x)-1\right )^{\frac {1}{4} i \left (-\sqrt {-\frac {1}{(a-1) a}-4}-\frac {i}{\sqrt {a-1} \sqrt {a}}\right ) \sqrt {a-1} \sqrt {a}}+2 y \left (\cos ^2(x)-1\right )^{\frac {1}{4} i \left (-\sqrt {-\frac {1}{(a-1) a}-4}-\frac {i}{\sqrt {a-1} \sqrt {a}}\right ) \sqrt {a-1} \sqrt {a}}-i \sqrt {-\frac {1}{(a-1) a}-4} \sqrt {a-1} \sqrt {a} \sin (x) \cos (x) \left (\cos ^2(x)-1\right )^{\frac {1}{4} i \left (-\sqrt {-\frac {1}{(a-1) a}-4}-\frac {i}{\sqrt {a-1} \sqrt {a}}\right ) \sqrt {a-1} \sqrt {a}}+\sin (x) \cos (x) \left (\cos ^2(x)-1\right )^{\frac {1}{4} i \left (-\sqrt {-\frac {1}{(a-1) a}-4}-\frac {i}{\sqrt {a-1} \sqrt {a}}\right ) \sqrt {a-1} \sqrt {a}}}\right )\right \}\right \} \]
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';v:='v'; pde := diff(w(x,y),x)+ (y^2-y *tan(x)+a*(1-a)*cot(x)^2)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac { \left ( \sin \left ( x \right ) \right ) ^{2\,a-1} \left ( y\sin \left ( x \right ) +\cos \left ( x \right ) a \right ) }{y\sin \left ( x \right ) -\cos \left ( x \right ) a+\cos \left ( x \right ) }} \right ) \]
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Added January 20, 2019.
Problem 2.6.5.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + \left ( y^2-m y \tan x+b^2 \cos ^{2 m} x \right ) w_y = 0 \]
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]; pde = D[w[x, y], x] + (y^2 - m*y*Tan[x] + b^2*Cos[x]^(2*m))*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to c_1\left (\frac {\sqrt {b^2} \sqrt {\sin ^2(x)} \csc (x) \cos ^{m+1}(x) \text {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},\cos ^2(x)\right )}{m+1}+\tan ^{-1}\left (\frac {y \cos ^{-m}(x)}{\sqrt {b^2}}\right )\right )\right \}\right \} \]
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';v:='v'; pde := diff(w(x,y),x)+ (y^2-m*y*tan(x)+b^2*cos(x)^(2*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 ( {\frac { \left ( \cos \left ( x \right ) \right ) ^{4}\sqrt { \left ( \cos \left ( x \right ) \right ) ^{2\,m-2}}{\mbox {$_2$F$_1$}(3/2,-m/2+3/2;\,5/2;\, \left ( \sin \left ( x \right ) \right ) ^{2})}\cos \left ( b\sqrt { \left ( \cos \left ( x \right ) \right ) ^{2\,m-2}} \left ( \cos \left ( x \right ) \right ) ^{1-m}\sin \left ( x \right ) {\mbox {$_2$F$_1$}(1/2,-m/2+1/2;\,3/2;\, \left ( \sin \left ( x \right ) \right ) ^{2})} \right ) bm- \left ( \cos \left ( x \right ) \right ) ^{4}\sqrt { \left ( \cos \left ( x \right ) \right ) ^{2\,m-2}}{\mbox {$_2$F$_1$}(3/2,-m/2+3/2;\,5/2;\, \left ( \sin \left ( x \right ) \right ) ^{2})}\cos \left ( b\sqrt { \left ( \cos \left ( x \right ) \right ) ^{2\,m-2}} \left ( \cos \left ( x \right ) \right ) ^{1-m}\sin \left ( x \right ) {\mbox {$_2$F$_1$}(1/2,-m/2+1/2;\,3/2;\, \left ( \sin \left ( x \right ) \right ) ^{2})} \right ) b- \left ( \cos \left ( x \right ) \right ) ^{2}\sqrt { \left ( \cos \left ( x \right ) \right ) ^{2\,m-2}}\cos \left ( b\sqrt { \left ( \cos \left ( x \right ) \right ) ^{2\,m-2}} \left ( \cos \left ( x \right ) \right ) ^{1-m}\sin \left ( x \right ) {\mbox {$_2$F$_1$}(1/2,-m/2+1/2;\,3/2;\, \left ( \sin \left ( x \right ) \right ) ^{2})} \right ) {\mbox {$_2$F$_1$}(3/2,-m/2+3/2;\,5/2;\, \left ( \sin \left ( x \right ) \right ) ^{2})}bm+3\, \left ( \cos \left ( x \right ) \right ) ^{2}\sqrt { \left ( \cos \left ( x \right ) \right ) ^{2\,m-2}}\cos \left ( b\sqrt { \left ( \cos \left ( x \right ) \right ) ^{2\,m-2}} \left ( \cos \left ( x \right ) \right ) ^{1-m}\sin \left ( x \right ) {\mbox {$_2$F$_1$}(1/2,-m/2+1/2;\,3/2;\, \left ( \sin \left ( x \right ) \right ) ^{2})} \right ) {\mbox {$_2$F$_1$}(1/2,-m/2+1/2;\,3/2;\, \left ( \sin \left ( x \right ) \right ) ^{2})}b+ \left ( \cos \left ( x \right ) \right ) ^{2}\sqrt { \left ( \cos \left ( x \right ) \right ) ^{2\,m-2}}\cos \left ( b\sqrt { \left ( \cos \left ( x \right ) \right ) ^{2\,m-2}} \left ( \cos \left ( x \right ) \right ) ^{1-m}\sin \left ( x \right ) {\mbox {$_2$F$_1$}(1/2,-m/2+1/2;\,3/2;\, \left ( \sin \left ( x \right ) \right ) ^{2})} \right ) {\mbox {$_2$F$_1$}(3/2,-m/2+3/2;\,5/2;\, \left ( \sin \left ( x \right ) \right ) ^{2})}b+3\,y\sin \left ( b\sqrt { \left ( \cos \left ( x \right ) \right ) ^{2\,m-2}} \left ( \cos \left ( x \right ) \right ) ^{1-m}\sin \left ( x \right ) {\mbox {$_2$F$_1$}(1/2,-m/2+1/2;\,3/2;\, \left ( \sin \left ( x \right ) \right ) ^{2})} \right ) \left ( \cos \left ( x \right ) \right ) ^{m}}{ \left ( \cos \left ( x \right ) \right ) ^{4}\sqrt { \left ( \cos \left ( x \right ) \right ) ^{2\,m-2}}{\mbox {$_2$F$_1$}(3/2,-m/2+3/2;\,5/2;\, \left ( \sin \left ( x \right ) \right ) ^{2})}\sin \left ( b\sqrt { \left ( \cos \left ( x \right ) \right ) ^{2\,m-2}} \left ( \cos \left ( x \right ) \right ) ^{1-m}\sin \left ( x \right ) {\mbox {$_2$F$_1$}(1/2,-m/2+1/2;\,3/2;\, \left ( \sin \left ( x \right ) \right ) ^{2})} \right ) bm- \left ( \cos \left ( x \right ) \right ) ^{4}\sqrt { \left ( \cos \left ( x \right ) \right ) ^{2\,m-2}}{\mbox {$_2$F$_1$}(3/2,-m/2+3/2;\,5/2;\, \left ( \sin \left ( x \right ) \right ) ^{2})}\sin \left ( b\sqrt { \left ( \cos \left ( x \right ) \right ) ^{2\,m-2}} \left ( \cos \left ( x \right ) \right ) ^{1-m}\sin \left ( x \right ) {\mbox {$_2$F$_1$}(1/2,-m/2+1/2;\,3/2;\, \left ( \sin \left ( x \right ) \right ) ^{2})} \right ) b- \left ( \cos \left ( x \right ) \right ) ^{2}\sqrt { \left ( \cos \left ( x \right ) \right ) ^{2\,m-2}}\sin \left ( b\sqrt { \left ( \cos \left ( x \right ) \right ) ^{2\,m-2}} \left ( \cos \left ( x \right ) \right ) ^{1-m}\sin \left ( x \right ) {\mbox {$_2$F$_1$}(1/2,-m/2+1/2;\,3/2;\, \left ( \sin \left ( x \right ) \right ) ^{2})} \right ) {\mbox {$_2$F$_1$}(3/2,-m/2+3/2;\,5/2;\, \left ( \sin \left ( x \right ) \right ) ^{2})}bm+3\, \left ( \cos \left ( x \right ) \right ) ^{2}\sqrt { \left ( \cos \left ( x \right ) \right ) ^{2\,m-2}}\sin \left ( b\sqrt { \left ( \cos \left ( x \right ) \right ) ^{2\,m-2}} \left ( \cos \left ( x \right ) \right ) ^{1-m}\sin \left ( x \right ) {\mbox {$_2$F$_1$}(1/2,-m/2+1/2;\,3/2;\, \left ( \sin \left ( x \right ) \right ) ^{2})} \right ) {\mbox {$_2$F$_1$}(1/2,-m/2+1/2;\,3/2;\, \left ( \sin \left ( x \right ) \right ) ^{2})}b+ \left ( \cos \left ( x \right ) \right ) ^{2}\sqrt { \left ( \cos \left ( x \right ) \right ) ^{2\,m-2}}\sin \left ( b\sqrt { \left ( \cos \left ( x \right ) \right ) ^{2\,m-2}} \left ( \cos \left ( x \right ) \right ) ^{1-m}\sin \left ( x \right ) {\mbox {$_2$F$_1$}(1/2,-m/2+1/2;\,3/2;\, \left ( \sin \left ( x \right ) \right ) ^{2})} \right ) {\mbox {$_2$F$_1$}(3/2,-m/2+3/2;\,5/2;\, \left ( \sin \left ( x \right ) \right ) ^{2})}b-3\,y\cos \left ( b\sqrt { \left ( \cos \left ( x \right ) \right ) ^{2\,m-2}} \left ( \cos \left ( x \right ) \right ) ^{1-m}\sin \left ( x \right ) {\mbox {$_2$F$_1$}(1/2,-m/2+1/2;\,3/2;\, \left ( \sin \left ( x \right ) \right ) ^{2})} \right ) \left ( \cos \left ( x \right ) \right ) ^{m}}} \right ) \] Mathematica answer is simpler
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Added January 20, 2019.
Problem 2.6.5.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + \left ( y^2+m y \cot x+b^2 \sin ^m x \right ) w_y = 0 \]
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]; pde = D[w[x, y], x] + (y^2 + m*y*Cot[x] + b^2*Sin[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';v:='v'; pde := diff(w(x,y),x)+ (y^2+m*y*cot(x)+b^2*sin(x)^m )*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ \text { Exception } \] Server hangs
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Added January 20, 2019.
Problem 2.6.5.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + \left ( y^2-2 \lambda ^2 \tan ^2(\lambda x)-2 \lambda ^2 \cot ^2(\lambda x) \right ) w_y = 0 \]
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]; pde = D[w[x, y], x] + (y^2 - 2*lambda^2*Tan[lambda*x]^2 - 2*lambda^2*Cot[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';v:='v'; pde := diff(w(x,y),x)+ (y^2-2*lambda^2*tan(lambda*x)^2-2*lambda^2*cot(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'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( -8\,{\frac {\sqrt {-2+2\,\cos \left ( 2\,\lambda \,x \right ) } \left ( \sin \left ( 2\,\lambda \,x \right ) y-2\,\cos \left ( 2\,\lambda \,x \right ) \lambda \right ) }{8\,\sqrt {-2+2\,\cos \left ( 2\,\lambda \,x \right ) }\ln \left ( \cos \left ( \lambda \,x \right ) +1/2\,\sqrt {-2+2\,\cos \left ( 2\,\lambda \,x \right ) } \right ) \sin \left ( 2\,\lambda \,x \right ) y-16\,\sqrt {-2+2\,\cos \left ( 2\,\lambda \,x \right ) }\ln \left ( \cos \left ( \lambda \,x \right ) +1/2\,\sqrt {-2+2\,\cos \left ( 2\,\lambda \,x \right ) } \right ) \cos \left ( 2\,\lambda \,x \right ) \lambda -4\,\sin \left ( \lambda \,x \right ) \left ( \cos \left ( 2\,\lambda \,x \right ) \right ) ^{2}y+14\,\cos \left ( 3\,\lambda \,x \right ) \lambda -2\,\cos \left ( 7\,\lambda \,x \right ) \lambda +2\,\cos \left ( 5\,\lambda \,x \right ) \lambda +2\,\sin \left ( 5\,\lambda \,x \right ) y-\sin \left ( 7\,\lambda \,x \right ) y+\sin \left ( \lambda \,x \right ) y-14\,\lambda \,\cos \left ( \lambda \,x \right ) }} \right ) \]
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Added January 20, 2019.
Problem 2.6.5.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + \left ( y^2+\lambda (a+b)+2 a b+a(\lambda -a) \tan ^2(\lambda x)+ b(\lambda -b) \cot ^2(\lambda x) \right ) w_y = 0 \]
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]; pde = D[w[x, y], x] + (y^2 + lambda*(a + b) + 2*a*b + a*(lambda - a)*Tan[lambda*x]^2 + b*(lambda - b)*Cot[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';v:='v'; pde := diff(w(x,y),x)+ ( y^2+lambda*(a+b)+2*a*b+a*(lambda -a)*tan(lambda*x)^2+ b*(lambda -b)*cot(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 ( {(2\,{a}^{2} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}-3\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}a\lambda -2\,\sin \left ( \lambda \,x \right ) \cos \left ( \lambda \,x \right ) ya+3\,\sin \left ( \lambda \,x \right ) \cos \left ( \lambda \,x \right ) y\lambda -2\, \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}ab+3\, \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}b\lambda ) \left ( \cos \left ( \lambda \,x \right ) \right ) ^{{\frac {a}{\lambda }}} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{{\frac {b}{\lambda }}} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{{\frac {a-\lambda }{\lambda }}} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{{\frac {b-\lambda }{\lambda }}} \left ( -4\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}{\mbox {$_2$F$_1$}(2,-{\frac {-2\,\lambda +b+a}{\lambda }};\,-1/2\,{\frac {2\,a-5\,\lambda }{\lambda }};\, \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2})} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}a\lambda -4\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}{\mbox {$_2$F$_1$}(2,-{\frac {-2\,\lambda +b+a}{\lambda }};\,-1/2\,{\frac {2\,a-5\,\lambda }{\lambda }};\, \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2})} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}b\lambda +4\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}{\mbox {$_2$F$_1$}(2,-{\frac {-2\,\lambda +b+a}{\lambda }};\,-1/2\,{\frac {2\,a-5\,\lambda }{\lambda }};\, \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2})} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}{\lambda }^{2}+2\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}{\mbox {$_2$F$_1$}(1,-{\frac {b-\lambda +a}{\lambda }};\,-1/2\,{\frac {2\,a-3\,\lambda }{\lambda }};\, \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2})}{a}^{2}-5\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}{\mbox {$_2$F$_1$}(1,-{\frac {b-\lambda +a}{\lambda }};\,-1/2\,{\frac {2\,a-3\,\lambda }{\lambda }};\, \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2})}a\lambda +3\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}{\mbox {$_2$F$_1$}(1,-{\frac {b-\lambda +a}{\lambda }};\,-1/2\,{\frac {2\,a-3\,\lambda }{\lambda }};\, \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2})}{\lambda }^{2}+2\,\sin \left ( \lambda \,x \right ) {\mbox {$_2$F$_1$}(1,-{\frac {b-\lambda +a}{\lambda }};\,-1/2\,{\frac {2\,a-3\,\lambda }{\lambda }};\, \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2})}\cos \left ( \lambda \,x \right ) ya-3\,\sin \left ( \lambda \,x \right ) {\mbox {$_2$F$_1$}(1,-{\frac {b-\lambda +a}{\lambda }};\,-1/2\,{\frac {2\,a-3\,\lambda }{\lambda }};\, \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2})}\cos \left ( \lambda \,x \right ) y\lambda -2\,{\mbox {$_2$F$_1$}(1,-{\frac {b-\lambda +a}{\lambda }};\,-1/2\,{\frac {2\,a-3\,\lambda }{\lambda }};\, \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2})} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}ab+2\,{\mbox {$_2$F$_1$}(1,-{\frac {b-\lambda +a}{\lambda }};\,-1/2\,{\frac {2\,a-3\,\lambda }{\lambda }};\, \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2})} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}a\lambda +3\,{\mbox {$_2$F$_1$}(1,-{\frac {b-\lambda +a}{\lambda }};\,-1/2\,{\frac {2\,a-3\,\lambda }{\lambda }};\, \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2})} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}b\lambda -3\,{\mbox {$_2$F$_1$}(1,-{\frac {b-\lambda +a}{\lambda }};\,-1/2\,{\frac {2\,a-3\,\lambda }{\lambda }};\, \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2})} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}{\lambda }^{2} \right ) ^{-1}} \right ) \]
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Added January 20, 2019.
Problem 2.6.5.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + \left ( \lambda \sin (\lambda x) y^2 + a \cos ^n(\lambda x) y-a \cos ^{n-1}(\lambda x) \right ) w_y = 0 \]
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]; pde = D[w[x, y], x] + (lambda*Sin[lambda*x]*y^2 + a*Cos[lambda*x]^n*y - a*Cos[lambda*x]^(n - 1))*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';v:='v'; pde := diff(w(x,y),x)+ (lambda*sin(lambda*x)* y^2 + a*cos(lambda*x)^n*y-a*cos(lambda*x)^(n-1))*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 2.6.5.8 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + \left ( \lambda \sin (\lambda x) y^2 + a \sin (\lambda x) y-a \tan (\lambda x) \right ) w_y = 0 \]
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]; pde = D[w[x, y], x] + (lambda*Sin[lambda*x]*y^2 + a*Sin[lambda*x]*y - a*Tan[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';v:='v'; pde := diff(w(x,y),x)+ (lambda*sin(lambda*x)*y^2 + a*sin(lambda*x)*y-a*tan(lambda*x))*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{(y\cos \left ( \lambda \,x \right ) -1){{\rm e}^{{\frac {\cos \left ( \lambda \,x \right ) a}{\lambda }}}} \left ( \cos \left ( \lambda \,x \right ) y\Ei \left ( 1,{\frac {\cos \left ( \lambda \,x \right ) a}{\lambda }} \right ) {{\rm e}^{{\frac {\cos \left ( \lambda \,x \right ) a}{\lambda }}}}a-\Ei \left ( 1,{\frac {\cos \left ( \lambda \,x \right ) a}{\lambda }} \right ) {{\rm e}^{{\frac {\cos \left ( \lambda \,x \right ) a}{\lambda }}}}a-y\lambda \right ) ^{-1}} \right ) \]
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Added January 20, 2019.
Problem 2.6.5.9 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + \left ( \lambda \sin (\lambda x) y^2 + a \sin (\lambda x) y-a \tan (\lambda x) \right ) w_y = 0 \]
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]; pde = D[w[x, y], x] + (lambda*Sin[lambda*x]*y^2 + a*Sin[lambda*x]*y - a*Tan[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';v:='v'; pde := diff(w(x,y),x)+ (lambda*sin(lambda*x)*y^2 + a*sin(lambda*x)*y-a*tan(lambda*x))*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{(y\cos \left ( \lambda \,x \right ) -1){{\rm e}^{{\frac {\cos \left ( \lambda \,x \right ) a}{\lambda }}}} \left ( \cos \left ( \lambda \,x \right ) y\Ei \left ( 1,{\frac {\cos \left ( \lambda \,x \right ) a}{\lambda }} \right ) {{\rm e}^{{\frac {\cos \left ( \lambda \,x \right ) a}{\lambda }}}}a-\Ei \left ( 1,{\frac {\cos \left ( \lambda \,x \right ) a}{\lambda }} \right ) {{\rm e}^{{\frac {\cos \left ( \lambda \,x \right ) a}{\lambda }}}}a-y\lambda \right ) ^{-1}} \right ) \]
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Added January 20, 2019.
Problem 2.6.5.10 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + \left ( A e^{\lambda x} \cos (a y) + B e^{\mu x} \sin (a y) + A e^{\lambda x} \right ) w_y = 0 \]
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]; pde = D[w[x, y], x] + (A*Exp[lambda*x]*Cos[a*y] + B*Exp[mu*x]*Sin[a*y] + A*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'; C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v'; pde := diff(w(x,y),x)+ (A*exp(lambda*x)*cos(a*y) + B*exp(mu*x)*sin(a*y) + A*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'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( -1/2\,{\frac {1}{a \left ( \lambda -\mu \right ) \left ( \cos \left ( 1/2\,ya \right ) \right ) ^{3}} \left ( \int \!{{\rm e}^{-{\frac {B{{\rm e}^{\mu \,x}}a-\mu \,x\lambda }{\mu }}}}\,{\rm d}xa\cos \left ( ya \right ) A\cos \left ( 1/2\,ya \right ) -{{\rm e}^{-{\frac {B{{\rm e}^{\mu \,x}}a}{\mu }}}}\sin \left ( ya \right ) \cos \left ( 1/2\,ya \right ) +\int \!{{\rm e}^{-{\frac {B{{\rm e}^{\mu \,x}}a-\mu \,x\lambda }{\mu }}}}\,{\rm d}xaA\cos \left ( 1/2\,ya \right ) \right ) } \right ) \]
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Added January 20, 2019.
Problem 2.6.5.11 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ \sin ^{n+1}(2 x) w_x + \left ( a y^2 \sin ^{2 n}x + b \cos ^{2 n} x \right ) w_y = 0 \]
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]; pde = Sin[2*x]^(n + 1)*D[w[x, y], x] + (a*y^2*Sin[x]^(2*n) + b*Cos[x]^(2*n))*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';v:='v'; pde := sin(2*x)^(n+1)*diff(w(x,y),x)+ (a*y^2*sin(x)^(2*n) + b*cos(x)^(2*n) )*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ \text { sol=() } \]