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Added January 29, 2019.
Problem 2.7.2.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + \left ( a \arccos ^k(\lambda x) + b \right ) w_y = 0 \]
Mathematica ✓
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B, v]; pde = D[w[x, y], x] + (a*ArcCos[lambda*x]^k + b)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to c_1\left (-\frac {\left (\cos ^{-1}(\lambda x)^2\right )^{-k} \left (a \left (i \cos ^{-1}(\lambda x)\right )^k \cos ^{-1}(\lambda x)^k \text {Gamma}\left (k+1,-i \cos ^{-1}(\lambda x)\right )+a \left (-i \cos ^{-1}(\lambda x)\right )^k \cos ^{-1}(\lambda x)^k \text {Gamma}\left (k+1,i \cos ^{-1}(\lambda x)\right )+2 b \lambda x \left (\cos ^{-1}(\lambda x)^2\right )^k-2 \lambda y \left (\cos ^{-1}(\lambda x)^2\right )^k\right )}{2 \lambda }\right )\right \}\right \} \]
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*arccos(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'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( -bx+y+{\frac {a{2}^{k}\sqrt {\pi }}{\lambda } \left ( {\frac { \left ( \arccos \left ( \lambda \,x \right ) \right ) ^{k+1}{2}^{-k}\sqrt {-{\lambda }^{2}{x}^{2}+1}}{\sqrt {\pi } \left ( k+2 \right ) }}-{\frac {{2}^{-k}\sqrt {\arccos \left ( \lambda \,x \right ) }\LommelS 1 \left ( 3/2+k,3/2,\arccos \left ( \lambda \,x \right ) \right ) \sqrt {-{\lambda }^{2}{x}^{2}+1}}{\sqrt {\pi } \left ( k+2 \right ) }}-3\,{\frac {{2}^{-1-k} \left ( 2/3\,k+4/3 \right ) \left ( \arccos \left ( \lambda \,x \right ) x\lambda -\sqrt {-{\lambda }^{2}{x}^{2}+1} \right ) \LommelS 1 \left ( k+1/2,1/2,\arccos \left ( \lambda \,x \right ) \right ) }{\sqrt {\pi } \left ( k+2 \right ) \sqrt {\arccos \left ( \lambda \,x \right ) }}} \right ) } \right ) \]
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Added January 29, 2019.
Problem 2.7.2.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + \left ( a \arccos ^k(\lambda y) + b \right ) w_y = 0 \]
Mathematica ✗
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B, v]; pde = D[w[x, y], x] + (a*ArcCos[lambda*y]^k + b)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \text {\$Aborted} \]
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*arccos(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'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( -\int \! \left ( a \left ( \arccos \left ( y\lambda \right ) \right ) ^{k}+b \right ) ^{-1}\,{\rm d}y+x \right ) \]
____________________________________________________________________________________
Added January 29, 2019.
Problem 2.7.2.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + k \arccos ^n(a x+b y+c) w_y = 0 \]
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] + k*ArcCos[a*x + b*y + c]^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 := diff(w(x,y),x)+k*arccos(a*x+b*y+c)^n*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( -\int ^{{\frac {ax+by}{b}}}\! \left ( k \left ( \arccos \left ( b{\it \_a}+c \right ) \right ) ^{n}b+a \right ) ^{-1}{d{\it \_a}}b+x \right ) \]
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Added January 29, 2019.
Problem 2.7.2.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + a \arccos ^k(\lambda x) \arccos ^n(\mu y) w_y = 0 \]
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*ArcCos[lambda*x]^k*ArcCos[mu*y]^n*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to c_1\left (\frac {\left (\cos ^{-1}(\lambda x)^2\right )^{-k} \left (-a \left (i \cos ^{-1}(\lambda x)\right )^k \cos ^{-1}(\lambda x)^k \text {Gamma}\left (k+1,-i \cos ^{-1}(\lambda x)\right )-a \left (-i \cos ^{-1}(\lambda x)\right )^k \cos ^{-1}(\lambda x)^k \text {Gamma}\left (k+1,i \cos ^{-1}(\lambda x)\right )+\frac {\lambda \left (\cos ^{-1}(\lambda x)^2\right )^k \cos ^{-1}(\mu y)^{-n} \left (\left (-i \cos ^{-1}(\mu y)\right )^n \text {Gamma}\left (1-n,-i \cos ^{-1}(\mu y)\right )+\left (i \cos ^{-1}(\mu y)\right )^n \text {Gamma}\left (1-n,i \cos ^{-1}(\mu y)\right )\right )}{\mu }\right )}{2 \lambda }\right )\right \}\right \} \]
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*arccos(lambda*x)^k*arccos(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 {1}{ \left ( n-2 \right ) \mu \,a\sqrt {\arccos \left ( \mu \,y \right ) }} \left ( \arccos \left ( \mu \,y \right ) y\LommelS 1 \left ( -n+1/2,1/2,\arccos \left ( \mu \,y \right ) \right ) \mu \,n-2\,\arccos \left ( \mu \,y \right ) y\LommelS 1 \left ( -n+1/2,1/2,\arccos \left ( \mu \,y \right ) \right ) \mu +{\frac {a{2}^{k}\sqrt {\pi }\mu \,n\sqrt {\arccos \left ( \mu \,y \right ) }}{\lambda } \left ( {\frac { \left ( \arccos \left ( \lambda \,x \right ) \right ) ^{k+1}{2}^{-k}\sqrt {-{\lambda }^{2}{x}^{2}+1}}{\sqrt {\pi } \left ( k+2 \right ) }}-{\frac {{2}^{-k}\sqrt {\arccos \left ( \lambda \,x \right ) }\LommelS 1 \left ( 3/2+k,3/2,\arccos \left ( \lambda \,x \right ) \right ) \sqrt {-{\lambda }^{2}{x}^{2}+1}}{\sqrt {\pi } \left ( k+2 \right ) }}-3\,{\frac {{2}^{-1-k} \left ( 2/3\,k+4/3 \right ) \left ( \arccos \left ( \lambda \,x \right ) x\lambda -\sqrt {-{\lambda }^{2}{x}^{2}+1} \right ) \LommelS 1 \left ( k+1/2,1/2,\arccos \left ( \lambda \,x \right ) \right ) }{\sqrt {\pi } \left ( k+2 \right ) \sqrt {\arccos \left ( \lambda \,x \right ) }}} \right ) }-2\,{\frac {a{2}^{k}\sqrt {\pi }\mu \,\sqrt {\arccos \left ( \mu \,y \right ) }}{\lambda } \left ( {\frac { \left ( \arccos \left ( \lambda \,x \right ) \right ) ^{k+1}{2}^{-k}\sqrt {-{\lambda }^{2}{x}^{2}+1}}{\sqrt {\pi } \left ( k+2 \right ) }}-{\frac {{2}^{-k}\sqrt {\arccos \left ( \lambda \,x \right ) }\LommelS 1 \left ( 3/2+k,3/2,\arccos \left ( \lambda \,x \right ) \right ) \sqrt {-{\lambda }^{2}{x}^{2}+1}}{\sqrt {\pi } \left ( k+2 \right ) }}-3\,{\frac {{2}^{-1-k} \left ( 2/3\,k+4/3 \right ) \left ( \arccos \left ( \lambda \,x \right ) x\lambda -\sqrt {-{\lambda }^{2}{x}^{2}+1} \right ) \LommelS 1 \left ( k+1/2,1/2,\arccos \left ( \lambda \,x \right ) \right ) }{\sqrt {\pi } \left ( k+2 \right ) \sqrt {\arccos \left ( \lambda \,x \right ) }}} \right ) }-\arccos \left ( \mu \,y \right ) \LommelS 1 \left ( -n+3/2,3/2,\arccos \left ( \mu \,y \right ) \right ) \sqrt {-{\mu }^{2}{y}^{2}+1}+ \left ( \arccos \left ( \mu \,y \right ) \right ) ^{-n+3/2}\sqrt {-{\mu }^{2}{y}^{2}+1}-\sqrt {-{\mu }^{2}{y}^{2}+1}\LommelS 1 \left ( -n+1/2,1/2,\arccos \left ( \mu \,y \right ) \right ) n+2\,\sqrt {-{\mu }^{2}{y}^{2}+1}\LommelS 1 \left ( -n+1/2,1/2,\arccos \left ( \mu \,y \right ) \right ) \right ) } \right ) \]
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Added January 29, 2019.
Problem 2.7.2.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + \left ( y^2+\lambda (\arccos x)^n y- a^2 + a \lambda ( \arccos x)^n \right ) w_y = 0 \]
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*ArcCos[x]^n*y - a^2 + a*lambda*ArcCos[x]^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 := diff(w(x,y),x)+( y^2+lambda*arccos(x)^n*y- a^2 + a*lambda*arccos(x)^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 {1}{ \left ( n+2 \right ) \left ( y+a \right ) } \left ( \int \!{\frac {\sqrt {\arccos \left ( x \right ) }\LommelS 1 \left ( -1/2+n,1/2,\arccos \left ( x \right ) \right ) x}{\sqrt {-{x}^{2}+1}}{{\rm e}^{-{\frac {-\arccos \left ( x \right ) \LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) xn\lambda -\sqrt {-{x}^{2}+1}\LommelS 1 \left ( n+3/2,3/2,\arccos \left ( x \right ) \right ) \arccos \left ( x \right ) \lambda +\sqrt {-{x}^{2}+1}\LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) n\lambda -2\,\arccos \left ( x \right ) \LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) x\lambda +\sqrt {-{x}^{2}+1} \left ( \arccos \left ( x \right ) \right ) ^{n+3/2}\lambda +2\,\sqrt {-{x}^{2}+1}\LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) \lambda +2\,ax\sqrt {\arccos \left ( x \right ) }n+4\,ax\sqrt {\arccos \left ( x \right ) }}{ \left ( n+2 \right ) \sqrt {\arccos \left ( x \right ) }}}}}}\,{\rm d}x\lambda \,{n}^{2}-\int \!{\frac {\LommelS 1 \left ( -1/2+n,1/2,\arccos \left ( x \right ) \right ) }{\sqrt {\arccos \left ( x \right ) }}{{\rm e}^{-{\frac {-\arccos \left ( x \right ) \LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) xn\lambda -\sqrt {-{x}^{2}+1}\LommelS 1 \left ( n+3/2,3/2,\arccos \left ( x \right ) \right ) \arccos \left ( x \right ) \lambda +\sqrt {-{x}^{2}+1}\LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) n\lambda -2\,\arccos \left ( x \right ) \LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) x\lambda +\sqrt {-{x}^{2}+1} \left ( \arccos \left ( x \right ) \right ) ^{n+3/2}\lambda +2\,\sqrt {-{x}^{2}+1}\LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) \lambda +2\,ax\sqrt {\arccos \left ( x \right ) }n+4\,ax\sqrt {\arccos \left ( x \right ) }}{ \left ( n+2 \right ) \sqrt {\arccos \left ( x \right ) }}}}}}\,{\rm d}x\lambda \,{n}^{2}+2\,\int \!{\frac {\sqrt {\arccos \left ( x \right ) }\LommelS 1 \left ( -1/2+n,1/2,\arccos \left ( x \right ) \right ) x}{\sqrt {-{x}^{2}+1}}{{\rm e}^{-{\frac {-\arccos \left ( x \right ) \LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) xn\lambda -\sqrt {-{x}^{2}+1}\LommelS 1 \left ( n+3/2,3/2,\arccos \left ( x \right ) \right ) \arccos \left ( x \right ) \lambda +\sqrt {-{x}^{2}+1}\LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) n\lambda -2\,\arccos \left ( x \right ) \LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) x\lambda +\sqrt {-{x}^{2}+1} \left ( \arccos \left ( x \right ) \right ) ^{n+3/2}\lambda +2\,\sqrt {-{x}^{2}+1}\LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) \lambda +2\,ax\sqrt {\arccos \left ( x \right ) }n+4\,ax\sqrt {\arccos \left ( x \right ) }}{ \left ( n+2 \right ) \sqrt {\arccos \left ( x \right ) }}}}}}\,{\rm d}x\lambda \,n+\int \!{\frac {\LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) }{ \left ( \arccos \left ( x \right ) \right ) ^{3/2}}{{\rm e}^{-{\frac {-\arccos \left ( x \right ) \LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) xn\lambda -\sqrt {-{x}^{2}+1}\LommelS 1 \left ( n+3/2,3/2,\arccos \left ( x \right ) \right ) \arccos \left ( x \right ) \lambda +\sqrt {-{x}^{2}+1}\LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) n\lambda -2\,\arccos \left ( x \right ) \LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) x\lambda +\sqrt {-{x}^{2}+1} \left ( \arccos \left ( x \right ) \right ) ^{n+3/2}\lambda +2\,\sqrt {-{x}^{2}+1}\LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) \lambda +2\,ax\sqrt {\arccos \left ( x \right ) }n+4\,ax\sqrt {\arccos \left ( x \right ) }}{ \left ( n+2 \right ) \sqrt {\arccos \left ( x \right ) }}}}}}\,{\rm d}x\lambda \,n+y\int \!{{\rm e}^{-{\frac {-\arccos \left ( x \right ) \LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) xn\lambda -\sqrt {-{x}^{2}+1}\LommelS 1 \left ( n+3/2,3/2,\arccos \left ( x \right ) \right ) \arccos \left ( x \right ) \lambda +\sqrt {-{x}^{2}+1}\LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) n\lambda -2\,\arccos \left ( x \right ) \LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) x\lambda +\sqrt {-{x}^{2}+1} \left ( \arccos \left ( x \right ) \right ) ^{n+3/2}\lambda +2\,\sqrt {-{x}^{2}+1}\LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) \lambda +2\,ax\sqrt {\arccos \left ( x \right ) }n+4\,ax\sqrt {\arccos \left ( x \right ) }}{ \left ( n+2 \right ) \sqrt {\arccos \left ( x \right ) }}}}}\,{\rm d}xn-\int \!{\frac {\LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) x}{\sqrt {\arccos \left ( x \right ) }\sqrt {-{x}^{2}+1}}{{\rm e}^{-{\frac {-\arccos \left ( x \right ) \LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) xn\lambda -\sqrt {-{x}^{2}+1}\LommelS 1 \left ( n+3/2,3/2,\arccos \left ( x \right ) \right ) \arccos \left ( x \right ) \lambda +\sqrt {-{x}^{2}+1}\LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) n\lambda -2\,\arccos \left ( x \right ) \LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) x\lambda +\sqrt {-{x}^{2}+1} \left ( \arccos \left ( x \right ) \right ) ^{n+3/2}\lambda +2\,\sqrt {-{x}^{2}+1}\LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) \lambda +2\,ax\sqrt {\arccos \left ( x \right ) }n+4\,ax\sqrt {\arccos \left ( x \right ) }}{ \left ( n+2 \right ) \sqrt {\arccos \left ( x \right ) }}}}}}\,{\rm d}x\lambda \,n-2\,\int \!{\frac {\LommelS 1 \left ( -1/2+n,1/2,\arccos \left ( x \right ) \right ) }{\sqrt {\arccos \left ( x \right ) }}{{\rm e}^{-{\frac {-\arccos \left ( x \right ) \LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) xn\lambda -\sqrt {-{x}^{2}+1}\LommelS 1 \left ( n+3/2,3/2,\arccos \left ( x \right ) \right ) \arccos \left ( x \right ) \lambda +\sqrt {-{x}^{2}+1}\LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) n\lambda -2\,\arccos \left ( x \right ) \LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) x\lambda +\sqrt {-{x}^{2}+1} \left ( \arccos \left ( x \right ) \right ) ^{n+3/2}\lambda +2\,\sqrt {-{x}^{2}+1}\LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) \lambda +2\,ax\sqrt {\arccos \left ( x \right ) }n+4\,ax\sqrt {\arccos \left ( x \right ) }}{ \left ( n+2 \right ) \sqrt {\arccos \left ( x \right ) }}}}}}\,{\rm d}x\lambda \,n+\int \!{{\rm e}^{-{\frac {-\arccos \left ( x \right ) \LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) xn\lambda -\sqrt {-{x}^{2}+1}\LommelS 1 \left ( n+3/2,3/2,\arccos \left ( x \right ) \right ) \arccos \left ( x \right ) \lambda +\sqrt {-{x}^{2}+1}\LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) n\lambda -2\,\arccos \left ( x \right ) \LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) x\lambda +\sqrt {-{x}^{2}+1} \left ( \arccos \left ( x \right ) \right ) ^{n+3/2}\lambda +2\,\sqrt {-{x}^{2}+1}\LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) \lambda +2\,ax\sqrt {\arccos \left ( x \right ) }n+4\,ax\sqrt {\arccos \left ( x \right ) }}{ \left ( n+2 \right ) \sqrt {\arccos \left ( x \right ) }}}}}\,{\rm d}xan+2\,\int \!{\frac {\LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) }{ \left ( \arccos \left ( x \right ) \right ) ^{3/2}}{{\rm e}^{-{\frac {-\arccos \left ( x \right ) \LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) xn\lambda -\sqrt {-{x}^{2}+1}\LommelS 1 \left ( n+3/2,3/2,\arccos \left ( x \right ) \right ) \arccos \left ( x \right ) \lambda +\sqrt {-{x}^{2}+1}\LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) n\lambda -2\,\arccos \left ( x \right ) \LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) x\lambda +\sqrt {-{x}^{2}+1} \left ( \arccos \left ( x \right ) \right ) ^{n+3/2}\lambda +2\,\sqrt {-{x}^{2}+1}\LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) \lambda +2\,ax\sqrt {\arccos \left ( x \right ) }n+4\,ax\sqrt {\arccos \left ( x \right ) }}{ \left ( n+2 \right ) \sqrt {\arccos \left ( x \right ) }}}}}}\,{\rm d}x\lambda +2\,y\int \!{{\rm e}^{-{\frac {-\arccos \left ( x \right ) \LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) xn\lambda -\sqrt {-{x}^{2}+1}\LommelS 1 \left ( n+3/2,3/2,\arccos \left ( x \right ) \right ) \arccos \left ( x \right ) \lambda +\sqrt {-{x}^{2}+1}\LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) n\lambda -2\,\arccos \left ( x \right ) \LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) x\lambda +\sqrt {-{x}^{2}+1} \left ( \arccos \left ( x \right ) \right ) ^{n+3/2}\lambda +2\,\sqrt {-{x}^{2}+1}\LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) \lambda +2\,ax\sqrt {\arccos \left ( x \right ) }n+4\,ax\sqrt {\arccos \left ( x \right ) }}{ \left ( n+2 \right ) \sqrt {\arccos \left ( x \right ) }}}}}\,{\rm d}x+\int \!{\frac {\sqrt {\arccos \left ( x \right ) }\LommelS 1 \left ( n+3/2,3/2,\arccos \left ( x \right ) \right ) x}{\sqrt {-{x}^{2}+1}}{{\rm e}^{-{\frac {-\arccos \left ( x \right ) \LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) xn\lambda -\sqrt {-{x}^{2}+1}\LommelS 1 \left ( n+3/2,3/2,\arccos \left ( x \right ) \right ) \arccos \left ( x \right ) \lambda +\sqrt {-{x}^{2}+1}\LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) n\lambda -2\,\arccos \left ( x \right ) \LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) x\lambda +\sqrt {-{x}^{2}+1} \left ( \arccos \left ( x \right ) \right ) ^{n+3/2}\lambda +2\,\sqrt {-{x}^{2}+1}\LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) \lambda +2\,ax\sqrt {\arccos \left ( x \right ) }n+4\,ax\sqrt {\arccos \left ( x \right ) }}{ \left ( n+2 \right ) \sqrt {\arccos \left ( x \right ) }}}}}}\,{\rm d}x\lambda +n{{\rm e}^{-{\frac {-\arccos \left ( x \right ) \LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) xn\lambda -\sqrt {-{x}^{2}+1}\LommelS 1 \left ( n+3/2,3/2,\arccos \left ( x \right ) \right ) \arccos \left ( x \right ) \lambda +\sqrt {-{x}^{2}+1}\LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) n\lambda -2\,\arccos \left ( x \right ) \LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) x\lambda +\sqrt {-{x}^{2}+1} \left ( \arccos \left ( x \right ) \right ) ^{n+3/2}\lambda +2\,\sqrt {-{x}^{2}+1}\LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) \lambda +2\,ax\sqrt {\arccos \left ( x \right ) }n+4\,ax\sqrt {\arccos \left ( x \right ) }}{ \left ( n+2 \right ) \sqrt {\arccos \left ( x \right ) }}}}}+\int \!{{\rm e}^{-{\frac {-\arccos \left ( x \right ) \LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) xn\lambda -\sqrt {-{x}^{2}+1}\LommelS 1 \left ( n+3/2,3/2,\arccos \left ( x \right ) \right ) \arccos \left ( x \right ) \lambda +\sqrt {-{x}^{2}+1}\LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) n\lambda -2\,\arccos \left ( x \right ) \LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) x\lambda +\sqrt {-{x}^{2}+1} \left ( \arccos \left ( x \right ) \right ) ^{n+3/2}\lambda +2\,\sqrt {-{x}^{2}+1}\LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) \lambda +2\,ax\sqrt {\arccos \left ( x \right ) }n+4\,ax\sqrt {\arccos \left ( x \right ) }}{ \left ( n+2 \right ) \sqrt {\arccos \left ( x \right ) }}}}} \left ( \arccos \left ( x \right ) \right ) ^{n}\,{\rm d}x\lambda -2\,\int \!{\frac {\LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) x}{\sqrt {\arccos \left ( x \right ) }\sqrt {-{x}^{2}+1}}{{\rm e}^{-{\frac {-\arccos \left ( x \right ) \LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) xn\lambda -\sqrt {-{x}^{2}+1}\LommelS 1 \left ( n+3/2,3/2,\arccos \left ( x \right ) \right ) \arccos \left ( x \right ) \lambda +\sqrt {-{x}^{2}+1}\LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) n\lambda -2\,\arccos \left ( x \right ) \LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) x\lambda +\sqrt {-{x}^{2}+1} \left ( \arccos \left ( x \right ) \right ) ^{n+3/2}\lambda +2\,\sqrt {-{x}^{2}+1}\LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) \lambda +2\,ax\sqrt {\arccos \left ( x \right ) }n+4\,ax\sqrt {\arccos \left ( x \right ) }}{ \left ( n+2 \right ) \sqrt {\arccos \left ( x \right ) }}}}}}\,{\rm d}x\lambda +2\,\int \!{{\rm e}^{-{\frac {-\arccos \left ( x \right ) \LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) xn\lambda -\sqrt {-{x}^{2}+1}\LommelS 1 \left ( n+3/2,3/2,\arccos \left ( x \right ) \right ) \arccos \left ( x \right ) \lambda +\sqrt {-{x}^{2}+1}\LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) n\lambda -2\,\arccos \left ( x \right ) \LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) x\lambda +\sqrt {-{x}^{2}+1} \left ( \arccos \left ( x \right ) \right ) ^{n+3/2}\lambda +2\,\sqrt {-{x}^{2}+1}\LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) \lambda +2\,ax\sqrt {\arccos \left ( x \right ) }n+4\,ax\sqrt {\arccos \left ( x \right ) }}{ \left ( n+2 \right ) \sqrt {\arccos \left ( x \right ) }}}}}\,{\rm d}xa-\int \!{\frac { \left ( \arccos \left ( x \right ) \right ) ^{n+1}x}{\sqrt {-{x}^{2}+1}}{{\rm e}^{-{\frac {-\arccos \left ( x \right ) \LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) xn\lambda -\sqrt {-{x}^{2}+1}\LommelS 1 \left ( n+3/2,3/2,\arccos \left ( x \right ) \right ) \arccos \left ( x \right ) \lambda +\sqrt {-{x}^{2}+1}\LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) n\lambda -2\,\arccos \left ( x \right ) \LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) x\lambda +\sqrt {-{x}^{2}+1} \left ( \arccos \left ( x \right ) \right ) ^{n+3/2}\lambda +2\,\sqrt {-{x}^{2}+1}\LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) \lambda +2\,ax\sqrt {\arccos \left ( x \right ) }n+4\,ax\sqrt {\arccos \left ( x \right ) }}{ \left ( n+2 \right ) \sqrt {\arccos \left ( x \right ) }}}}}}\,{\rm d}x\lambda -\int \!{\frac {\LommelS 1 \left ( n+3/2,3/2,\arccos \left ( x \right ) \right ) }{\sqrt {\arccos \left ( x \right ) }}{{\rm e}^{-{\frac {-\arccos \left ( x \right ) \LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) xn\lambda -\sqrt {-{x}^{2}+1}\LommelS 1 \left ( n+3/2,3/2,\arccos \left ( x \right ) \right ) \arccos \left ( x \right ) \lambda +\sqrt {-{x}^{2}+1}\LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) n\lambda -2\,\arccos \left ( x \right ) \LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) x\lambda +\sqrt {-{x}^{2}+1} \left ( \arccos \left ( x \right ) \right ) ^{n+3/2}\lambda +2\,\sqrt {-{x}^{2}+1}\LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) \lambda +2\,ax\sqrt {\arccos \left ( x \right ) }n+4\,ax\sqrt {\arccos \left ( x \right ) }}{ \left ( n+2 \right ) \sqrt {\arccos \left ( x \right ) }}}}}}\,{\rm d}x\lambda +2\,{{\rm e}^{-{\frac {-\arccos \left ( x \right ) \LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) xn\lambda -\sqrt {-{x}^{2}+1}\LommelS 1 \left ( n+3/2,3/2,\arccos \left ( x \right ) \right ) \arccos \left ( x \right ) \lambda +\sqrt {-{x}^{2}+1}\LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) n\lambda -2\,\arccos \left ( x \right ) \LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) x\lambda +\sqrt {-{x}^{2}+1} \left ( \arccos \left ( x \right ) \right ) ^{n+3/2}\lambda +2\,\sqrt {-{x}^{2}+1}\LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) \lambda +2\,ax\sqrt {\arccos \left ( x \right ) }n+4\,ax\sqrt {\arccos \left ( x \right ) }}{ \left ( n+2 \right ) \sqrt {\arccos \left ( x \right ) }}}}} \right ) } \right ) \]
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Added January 29, 2019.
Problem 2.7.2.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + \left ( y^2+\lambda x (\arccos x)^n y+ \lambda ( \arccos x)^n \right ) w_y = 0 \]
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*x*ArcCos[x]^n*y + a*lambda*ArcCos[x]^n)*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*x*arccos(x)^n*y + a*lambda*arccos(x)^n )*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ \text { sol=() } \]
____________________________________________________________________________________
Added January 29, 2019.
Problem 2.7.2.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x - \left ( (k+1)x^k y^2 -\lambda (\arccos x)^n (x^{k+1} y-1) \right ) w_y = 0 \]
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] - ((k + 1)*x^k*y^2 - lambda*ArcCos[x]^n*(x^(k + 1)*y - 1))*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)-( (k+1)*x^k*y^2 -lambda*arccos(x)^n*(x^(k+1)*y-1) )*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1}{{x}^{k+1}y-1} \left ( y{x}^{k+1}\int \!{\frac {{{\rm e}^{\lambda \,\int \!{x}^{k+1} \left ( \arccos \left ( x \right ) \right ) ^{n}\,{\rm d}x}}}{{x}^{k}{x}^{2}}}\,{\rm d}xk+y{x}^{k+1}\int \!{\frac {{{\rm e}^{\lambda \,\int \!{x}^{k+1} \left ( \arccos \left ( x \right ) \right ) ^{n}\,{\rm d}x}}}{{x}^{k}{x}^{2}}}\,{\rm d}x-{{\rm e}^{\int \!{\frac {{x}^{k+1} \left ( \arccos \left ( x \right ) \right ) ^{n}\lambda \,x-2\,k-2}{x}}\,{\rm d}x}}{x}^{k+1}-\int \!{\frac {{{\rm e}^{\lambda \,\int \!{x}^{k+1} \left ( \arccos \left ( x \right ) \right ) ^{n}\,{\rm d}x}}}{{x}^{k}{x}^{2}}}\,{\rm d}xk-\int \!{\frac {{{\rm e}^{\lambda \,\int \!{x}^{k+1} \left ( \arccos \left ( x \right ) \right ) ^{n}\,{\rm d}x}}}{{x}^{k}{x}^{2}}}\,{\rm d}x \right ) } \right ) \]
____________________________________________________________________________________
Added January 29, 2019.
Problem 2.7.2.8 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + \left ( \lambda (\arccos x)^n y^2+ a y+ a b - b^2 \lambda (\arccos x)^n \right ) w_y = 0 \]
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*ArcCos[x]^n*y^2 + a*y + a*b - b^2*lambda*ArcCos[x]^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 := diff(w(x,y),x)+( lambda*arccos(x)^n*y^2+ a*y+ a*b - b^2*lambda*arccos(x)^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 {1}{b+y} \left ( \int \!{\frac {\lambda \, \left ( -2\, \left ( \arccos \left ( x \right ) \right ) ^{3}\LommelS 1 \left ( -1/2+n,1/2,\arccos \left ( x \right ) \right ) xb{n}^{2}+2\,\sqrt {-{x}^{2}+1} \left ( \arccos \left ( x \right ) \right ) ^{2}\LommelS 1 \left ( -1/2+n,1/2,\arccos \left ( x \right ) \right ) b{n}^{2}-4\, \left ( \arccos \left ( x \right ) \right ) ^{3}\LommelS 1 \left ( -1/2+n,1/2,\arccos \left ( x \right ) \right ) xbn+4\,\sqrt {-{x}^{2}+1} \left ( \arccos \left ( x \right ) \right ) ^{2}\LommelS 1 \left ( -1/2+n,1/2,\arccos \left ( x \right ) \right ) bn+2\, \left ( \arccos \left ( x \right ) \right ) ^{2}\LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) xbn+\sqrt {-{x}^{2}+1} \left ( \arccos \left ( x \right ) \right ) ^{5/2} \left ( \arccos \left ( x \right ) \right ) ^{n}bn+2\,\sqrt {-{x}^{2}+1} \left ( \arccos \left ( x \right ) \right ) ^{5/2} \left ( \arccos \left ( x \right ) \right ) ^{n}y-2\,\LommelS 1 \left ( n+3/2,3/2,\arccos \left ( x \right ) \right ) \left ( \arccos \left ( x \right ) \right ) ^{3}xb+2\, \left ( \arccos \left ( x \right ) \right ) ^{7/2} \left ( \arccos \left ( x \right ) \right ) ^{n}xb-2\,\sqrt {-{x}^{2}+1}\arccos \left ( x \right ) \LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) bn+4\, \left ( \arccos \left ( x \right ) \right ) ^{2}\LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) xb+2\,\sqrt {-{x}^{2}+1}\LommelS 1 \left ( n+3/2,3/2,\arccos \left ( x \right ) \right ) \left ( \arccos \left ( x \right ) \right ) ^{2}b+\sqrt {-{x}^{2}+1} \left ( \arccos \left ( x \right ) \right ) ^{5/2} \left ( \arccos \left ( x \right ) \right ) ^{n}yn-4\,\sqrt {-{x}^{2}+1}\arccos \left ( x \right ) \LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) b \right ) }{\sqrt {-{x}^{2}+1} \left ( \arccos \left ( x \right ) \right ) ^{5/2} \left ( n+2 \right ) \left ( b+y \right ) }{{\rm e}^{{\frac {-2\,\arccos \left ( x \right ) \LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) xbn\lambda -2\,\sqrt {-{x}^{2}+1}\LommelS 1 \left ( n+3/2,3/2,\arccos \left ( x \right ) \right ) \arccos \left ( x \right ) b\lambda +2\,\sqrt {-{x}^{2}+1}\LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) bn\lambda -4\,\arccos \left ( x \right ) \LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) xb\lambda +2\,\sqrt {-{x}^{2}+1} \left ( \arccos \left ( x \right ) \right ) ^{n} \left ( \arccos \left ( x \right ) \right ) ^{3/2}b\lambda +4\,\sqrt {-{x}^{2}+1}\LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) b\lambda +ax\sqrt {\arccos \left ( x \right ) }n+2\,ax\sqrt {\arccos \left ( x \right ) }}{ \left ( n+2 \right ) \sqrt {\arccos \left ( x \right ) }}}}}}\,{\rm d}xy+\int \!{\frac {\lambda \, \left ( -2\, \left ( \arccos \left ( x \right ) \right ) ^{3}\LommelS 1 \left ( -1/2+n,1/2,\arccos \left ( x \right ) \right ) xb{n}^{2}+2\,\sqrt {-{x}^{2}+1} \left ( \arccos \left ( x \right ) \right ) ^{2}\LommelS 1 \left ( -1/2+n,1/2,\arccos \left ( x \right ) \right ) b{n}^{2}-4\, \left ( \arccos \left ( x \right ) \right ) ^{3}\LommelS 1 \left ( -1/2+n,1/2,\arccos \left ( x \right ) \right ) xbn+4\,\sqrt {-{x}^{2}+1} \left ( \arccos \left ( x \right ) \right ) ^{2}\LommelS 1 \left ( -1/2+n,1/2,\arccos \left ( x \right ) \right ) bn+2\, \left ( \arccos \left ( x \right ) \right ) ^{2}\LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) xbn+\sqrt {-{x}^{2}+1} \left ( \arccos \left ( x \right ) \right ) ^{5/2} \left ( \arccos \left ( x \right ) \right ) ^{n}bn+2\,\sqrt {-{x}^{2}+1} \left ( \arccos \left ( x \right ) \right ) ^{5/2} \left ( \arccos \left ( x \right ) \right ) ^{n}y-2\,\LommelS 1 \left ( n+3/2,3/2,\arccos \left ( x \right ) \right ) \left ( \arccos \left ( x \right ) \right ) ^{3}xb+2\, \left ( \arccos \left ( x \right ) \right ) ^{7/2} \left ( \arccos \left ( x \right ) \right ) ^{n}xb-2\,\sqrt {-{x}^{2}+1}\arccos \left ( x \right ) \LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) bn+4\, \left ( \arccos \left ( x \right ) \right ) ^{2}\LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) xb+2\,\sqrt {-{x}^{2}+1}\LommelS 1 \left ( n+3/2,3/2,\arccos \left ( x \right ) \right ) \left ( \arccos \left ( x \right ) \right ) ^{2}b+\sqrt {-{x}^{2}+1} \left ( \arccos \left ( x \right ) \right ) ^{5/2} \left ( \arccos \left ( x \right ) \right ) ^{n}yn-4\,\sqrt {-{x}^{2}+1}\arccos \left ( x \right ) \LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) b \right ) }{\sqrt {-{x}^{2}+1} \left ( \arccos \left ( x \right ) \right ) ^{5/2} \left ( n+2 \right ) \left ( b+y \right ) }{{\rm e}^{{\frac {-2\,\arccos \left ( x \right ) \LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) xbn\lambda -2\,\sqrt {-{x}^{2}+1}\LommelS 1 \left ( n+3/2,3/2,\arccos \left ( x \right ) \right ) \arccos \left ( x \right ) b\lambda +2\,\sqrt {-{x}^{2}+1}\LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) bn\lambda -4\,\arccos \left ( x \right ) \LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) xb\lambda +2\,\sqrt {-{x}^{2}+1} \left ( \arccos \left ( x \right ) \right ) ^{n} \left ( \arccos \left ( x \right ) \right ) ^{3/2}b\lambda +4\,\sqrt {-{x}^{2}+1}\LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) b\lambda +ax\sqrt {\arccos \left ( x \right ) }n+2\,ax\sqrt {\arccos \left ( x \right ) }}{ \left ( n+2 \right ) \sqrt {\arccos \left ( x \right ) }}}}}}\,{\rm d}xb+{{\rm e}^{-{\frac {-2\,\sqrt {-{x}^{2}+1} \left ( \arccos \left ( x \right ) \right ) ^{n} \left ( \arccos \left ( x \right ) \right ) ^{3/2}b\lambda +2\,\arccos \left ( x \right ) \LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) xbn\lambda +2\,\sqrt {-{x}^{2}+1}\LommelS 1 \left ( n+3/2,3/2,\arccos \left ( x \right ) \right ) \arccos \left ( x \right ) b\lambda -2\,\sqrt {-{x}^{2}+1}\LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) bn\lambda +4\,\arccos \left ( x \right ) \LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) xb\lambda -4\,\sqrt {-{x}^{2}+1}\LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) b\lambda -ax\sqrt {\arccos \left ( x \right ) }n-2\,ax\sqrt {\arccos \left ( x \right ) }}{ \left ( n+2 \right ) \sqrt {\arccos \left ( x \right ) }}}}} \right ) } \right ) \]
____________________________________________________________________________________
Added January 29, 2019.
Problem 2.7.2.9 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + \left ( \lambda (\arccos x)^n y^2- b \lambda x^m (\arccos x)^n y + b m x^{m-1} \right ) w_y = 0 \]
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*ArcCos[x]^n*y^2 - b*lambda*x^m*ArcCos[x]^n*y + b*m*x^(m - 1))*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*arccos(x)^n*y^2- b*lambda*x^m*arccos(x)^n*y + b*m*x^(m-1) )*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 January 29, 2019.
Problem 2.7.2.10 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + \left ( \lambda (\arccos x)^n y^2+ b m x^{m-1} - \lambda b^2 x^{2 m} (\arccos x)^n \right ) w_y = 0 \]
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*ArcCos[x]^n*y^2 + b*m*x^(m - 1) - lambda*b^2*x^(2*m)*ArcCos[x]^n)*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*arccos(x)^n*y^2+ b*m*x^(m-1) - lambda*b^2*x^(2*m)*arccos(x)^n )*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ \text { sol=() } \]
____________________________________________________________________________________
Added January 29, 2019.
Problem 2.7.2.11 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + \left ( \lambda (\arccos x)^n (y- a x^m-b)^2 + a m x^{m-1} \right ) w_y = 0 \]
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*ArcCos[x]^n*(y - a*x^m - b)^2 + a*m*x^(m - 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*arccos(x)^n*(y- a*x^m-b)^2 + a*m*x^(m-1) )*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 {-\sqrt {-{x}^{2}+1}{x}^{m} \left ( \arccos \left ( x \right ) \right ) ^{3/2} \left ( \arccos \left ( x \right ) \right ) ^{n}a\lambda -\sqrt {-{x}^{2}+1} \left ( \arccos \left ( x \right ) \right ) ^{n} \left ( \arccos \left ( x \right ) \right ) ^{3/2}b\lambda +y\sqrt {-{x}^{2}+1} \left ( \arccos \left ( x \right ) \right ) ^{3/2} \left ( \arccos \left ( x \right ) \right ) ^{n}\lambda +{x}^{m}\arccos \left ( x \right ) \LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) a\lambda \,nx+\sqrt {-{x}^{2}+1}{x}^{m}\arccos \left ( x \right ) \LommelS 1 \left ( n+3/2,3/2,\arccos \left ( x \right ) \right ) a\lambda -\sqrt {-{x}^{2}+1}{x}^{m}\LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) a\lambda \,n+2\,{x}^{m}\arccos \left ( x \right ) \LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) a\lambda \,x+\arccos \left ( x \right ) \LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) xbn\lambda -y\arccos \left ( x \right ) \LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) \lambda \,nx-2\,\sqrt {-{x}^{2}+1}{x}^{m}\LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) a\lambda +\sqrt {-{x}^{2}+1}\LommelS 1 \left ( n+3/2,3/2,\arccos \left ( x \right ) \right ) \arccos \left ( x \right ) b\lambda -y\sqrt {-{x}^{2}+1}\arccos \left ( x \right ) \LommelS 1 \left ( n+3/2,3/2,\arccos \left ( x \right ) \right ) \lambda -\sqrt {-{x}^{2}+1}\LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) bn\lambda +y\sqrt {-{x}^{2}+1}\LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) \lambda \,n+2\,\arccos \left ( x \right ) \LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) xb\lambda -2\,y\arccos \left ( x \right ) \LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) \lambda \,x-2\,\sqrt {-{x}^{2}+1}\LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) b\lambda +2\,y\sqrt {-{x}^{2}+1}\LommelS 1 \left ( n+1/2,1/2,\arccos \left ( x \right ) \right ) \lambda -\sqrt {\arccos \left ( x \right ) }n-2\,\sqrt {\arccos \left ( x \right ) }}{\sqrt {\arccos \left ( x \right ) } \left ( {x}^{m}an+2\,a{x}^{m}+bn-yn+2\,b-2\,y \right ) }} \right ) \]
____________________________________________________________________________________
Added January 29, 2019.
Problem 2.7.2.12 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ x w_x + \left ( \lambda (\arccos x)^n y^2+ k y + \lambda b^2 x^{2 k} (\arccos x)^n \right ) w_y = 0 \]
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 = x*D[w[x, y], x] + (lambda*ArcCos[x]^n*y^2 + k*y + lambda*b^2*x^(2*k)*ArcCos[x]^n)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to c_1\left (\tan ^{-1}\left (\frac {y x^{-k}}{\sqrt {b^2}}\right )-\sqrt {b^2} \int _1^x \lambda K[1]^{k-1} \cos ^{-1}(K[1])^n \, dK[1]\right )\right \}\right \} \]
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';v:='v'; pde := x*diff(w(x,y),x)+( lambda*arccos(x)^n*y^2+ k*y + lambda*b^2*x^(2*k)*arccos(x)^n )*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( \lambda \,b\int \! \left ( \arccos \left ( x \right ) \right ) ^{n}{x}^{k-1}\,{\rm d}x-\arctan \left ( {\frac {{x}^{-k}y}{b}} \right ) \right ) \]