____________________________________________________________________________________
Added January 20, 2019.
Problem 2.7.1.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + \left ( a \arcsin ^k(\lambda x)+b \right ) w_y = 0 \]
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*ArcSin[lambda*x]^k + b)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to c_1\left (\frac {i a \sin ^{-1}(\lambda x)^k \left (\sin ^{-1}(\lambda x)^2\right )^{-k} \left (\left (i \sin ^{-1}(\lambda x)\right )^k \text {Gamma}\left (k+1,-i \sin ^{-1}(\lambda x)\right )-\left (-i \sin ^{-1}(\lambda x)\right )^k \text {Gamma}\left (k+1,i \sin ^{-1}(\lambda x)\right )\right )}{2 \lambda }-b x+y\right )\right \}\right \} \]
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';v:='v'; pde := diff(w(x,y),x)+ (a*arcsin(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'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {- \left ( \arcsin \left ( \lambda \,x \right ) \right ) ^{k} \left ( \arcsin \left ( \lambda \,x \right ) \right ) ^{3/2}{2}^{k}{2}^{-k}\sqrt {-{\lambda }^{2}{x}^{2}+1}a-\arcsin \left ( \lambda \,x \right ) {2}^{k}{2}^{-k}\LommelS 1 \left ( k+1/2,3/2,\arcsin \left ( \lambda \,x \right ) \right ) ak\lambda \,x-2\, \left ( \arcsin \left ( \lambda \,x \right ) \right ) ^{k}\sqrt {\arcsin \left ( \lambda \,x \right ) }{2}^{k}{2}^{-1-k}a\lambda \,x+ \left ( \arcsin \left ( \lambda \,x \right ) \right ) ^{k}\sqrt {\arcsin \left ( \lambda \,x \right ) }{2}^{k}{2}^{-k}a\lambda \,x+{2}^{k}{2}^{-k}\LommelS 1 \left ( 3/2+k,1/2,\arcsin \left ( \lambda \,x \right ) \right ) \sqrt {-{\lambda }^{2}{x}^{2}+1}\arcsin \left ( \lambda \,x \right ) a-{2}^{k}{2}^{-k}\LommelS 1 \left ( 3/2+k,1/2,\arcsin \left ( \lambda \,x \right ) \right ) a\lambda \,x-\sqrt {\arcsin \left ( \lambda \,x \right ) }bk\lambda \,x-bx\lambda \,\sqrt {\arcsin \left ( \lambda \,x \right ) }+\sqrt {\arcsin \left ( \lambda \,x \right ) }k\lambda \,y+y\lambda \,\sqrt {\arcsin \left ( \lambda \,x \right ) }}{\lambda \, \left ( k+1 \right ) \sqrt {\arcsin \left ( \lambda \,x \right ) }}} \right ) \]
____________________________________________________________________________________
Added January 20, 2019.
Problem 2.7.1.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + \left ( a \arcsin ^k(\lambda y)+b \right ) w_y = 0 \]
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*ArcSin[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*arcsin(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'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( -\int \! \left ( a \left ( \arcsin \left ( y\lambda \right ) \right ) ^{k}+b \right ) ^{-1}\,{\rm d}y+x \right ) \]
____________________________________________________________________________________
Added January 20, 2019.
Problem 2.7.1.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + k \arcsin ^n(a x + b y+c) w_y = 0 \]
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*Arcsin[a*x + b*y + c]^n*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \text {Failed} \]
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*arcsin(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'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( -\int ^{{\frac {ax+by}{b}}}\! \left ( k \left ( \arcsin \left ( b{\it \_a}+c \right ) \right ) ^{n}b+a \right ) ^{-1}{d{\it \_a}}b+x \right ) \]
____________________________________________________________________________________
Added January 20, 2019.
Problem 2.7.1.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + a \arcsin ^k(\lambda x) \arcsin ^n(\mu y) w_y = 0 \]
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*Arcsin[lambda*x]^k*Arcsin[mu*y]^n*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to c_1\left (\int _1^y \text {Arcsin}(\mu K[1])^{-n} \, dK[1]-\int _1^x a \text {Arcsin}(\lambda K[2])^k \, dK[2]\right )\right \}\right \} \]
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*arcsin(lambda*x)^k*arcsin(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 {{2}^{k}\sqrt {\pi }}{\lambda } \left ( {\frac {{2}^{-1-k} \left ( \arcsin \left ( \lambda \,x \right ) \right ) ^{k} \left ( 2\,k+6 \right ) x\lambda }{\sqrt {\pi } \left ( k+1 \right ) \left ( k+3 \right ) }}+{\frac { \left ( \arcsin \left ( \lambda \,x \right ) \right ) ^{k}{2}^{-k}\sqrt {-{\lambda }^{2}{x}^{2}+1} \left ( \arcsin \left ( \lambda \,x \right ) {x}^{2}{\lambda }^{2}-\arcsin \left ( \lambda \,x \right ) +\sqrt {-{\lambda }^{2}{x}^{2}+1}x\lambda \right ) }{\sqrt {\pi } \left ( k+1 \right ) \left ( {\lambda }^{2}{x}^{2}-1 \right ) }}+{\frac {{2}^{-k}\sqrt {\arcsin \left ( \lambda \,x \right ) }k\LommelS 1 \left ( k+1/2,3/2,\arcsin \left ( \lambda \,x \right ) \right ) x\lambda }{\sqrt {\pi } \left ( k+1 \right ) }}-{\frac {{2}^{-k}\sqrt {-{\lambda }^{2}{x}^{2}+1} \left ( \arcsin \left ( \lambda \,x \right ) {x}^{2}{\lambda }^{2}-\arcsin \left ( \lambda \,x \right ) +\sqrt {-{\lambda }^{2}{x}^{2}+1}x\lambda \right ) \LommelS 1 \left ( 3/2+k,1/2,\arcsin \left ( \lambda \,x \right ) \right ) }{\sqrt {\pi } \left ( k+1 \right ) \sqrt {\arcsin \left ( \lambda \,x \right ) } \left ( {\lambda }^{2}{x}^{2}-1 \right ) }} \right ) }+{\frac {{2}^{-n} \left ( {2}^{n}\arcsin \left ( \mu \,y \right ) n\LommelS 1 \left ( -n+1/2,3/2,\arcsin \left ( \mu \,y \right ) \right ) y\mu -{2}^{n} \left ( \arcsin \left ( \mu \,y \right ) \right ) ^{-n}\sqrt {-{\mu }^{2}{y}^{2}+1} \left ( \arcsin \left ( \mu \,y \right ) \right ) ^{3/2}+{2}^{n}y \left ( \arcsin \left ( \mu \,y \right ) \right ) ^{-n}\sqrt {\arcsin \left ( \mu \,y \right ) }\mu -2\,{2}^{n-1} \left ( \arcsin \left ( \mu \,y \right ) \right ) ^{-n}y\mu \,\sqrt {\arcsin \left ( \mu \,y \right ) }-{2}^{n}y\LommelS 1 \left ( -n+3/2,1/2,\arcsin \left ( \mu \,y \right ) \right ) \mu +{2}^{n}\sqrt {-{\mu }^{2}{y}^{2}+1}\arcsin \left ( \mu \,y \right ) \LommelS 1 \left ( -n+3/2,1/2,\arcsin \left ( \mu \,y \right ) \right ) \right ) }{ \left ( n-1 \right ) \mu \,a\sqrt {\arcsin \left ( \mu \,y \right ) }}} \right ) \]
____________________________________________________________________________________
Added January 20, 2019.
Problem 2.7.1.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + \left ( y^2+ \lambda (\arcsin x)^n y -a^2 + a \lambda ( \arcsin x)^n \right ) w_y = 0 \]
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*Arcsin[x]^n*y - a^2 + a*lambda*Arcsin[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*arcsin(x)^n*y -a^2 + a *lambda*arcsin(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}{y+a} \left ( y\int \!{{\rm e}^{\lambda \,{2}^{n}\sqrt {\pi } \left ( {\frac {{2}^{-n-1} \left ( \arcsin \left ( x \right ) \right ) ^{n} \left ( 2\,n+6 \right ) x}{\sqrt {\pi } \left ( n+1 \right ) \left ( n+3 \right ) }}+{\frac { \left ( \arcsin \left ( x \right ) \right ) ^{n}{2}^{-n}\sqrt {-{x}^{2}+1} \left ( \arcsin \left ( x \right ) {x}^{2}-\arcsin \left ( x \right ) +\sqrt {-{x}^{2}+1}x \right ) }{\sqrt {\pi } \left ( n+1 \right ) \left ( {x}^{2}-1 \right ) }}+{\frac {{2}^{-n}\sqrt {\arcsin \left ( x \right ) }n\LommelS 1 \left ( n+1/2,3/2,\arcsin \left ( x \right ) \right ) x}{\sqrt {\pi } \left ( n+1 \right ) }}-{\frac {{2}^{-n}\sqrt {-{x}^{2}+1} \left ( \arcsin \left ( x \right ) {x}^{2}-\arcsin \left ( x \right ) +\sqrt {-{x}^{2}+1}x \right ) \LommelS 1 \left ( n+3/2,1/2,\arcsin \left ( x \right ) \right ) }{\sqrt {\pi } \left ( n+1 \right ) \sqrt {\arcsin \left ( x \right ) } \left ( {x}^{2}-1 \right ) }} \right ) -2\,ax}}\,{\rm d}x+\int \!{{\rm e}^{\lambda \,{2}^{n}\sqrt {\pi } \left ( {\frac {{2}^{-n-1} \left ( \arcsin \left ( x \right ) \right ) ^{n} \left ( 2\,n+6 \right ) x}{\sqrt {\pi } \left ( n+1 \right ) \left ( n+3 \right ) }}+{\frac { \left ( \arcsin \left ( x \right ) \right ) ^{n}{2}^{-n}\sqrt {-{x}^{2}+1} \left ( \arcsin \left ( x \right ) {x}^{2}-\arcsin \left ( x \right ) +\sqrt {-{x}^{2}+1}x \right ) }{\sqrt {\pi } \left ( n+1 \right ) \left ( {x}^{2}-1 \right ) }}+{\frac {{2}^{-n}\sqrt {\arcsin \left ( x \right ) }n\LommelS 1 \left ( n+1/2,3/2,\arcsin \left ( x \right ) \right ) x}{\sqrt {\pi } \left ( n+1 \right ) }}-{\frac {{2}^{-n}\sqrt {-{x}^{2}+1} \left ( \arcsin \left ( x \right ) {x}^{2}-\arcsin \left ( x \right ) +\sqrt {-{x}^{2}+1}x \right ) \LommelS 1 \left ( n+3/2,1/2,\arcsin \left ( x \right ) \right ) }{\sqrt {\pi } \left ( n+1 \right ) \sqrt {\arcsin \left ( x \right ) } \left ( {x}^{2}-1 \right ) }} \right ) -2\,ax}}\,{\rm d}xa+{{\rm e}^{\lambda \,{2}^{n}\sqrt {\pi } \left ( {\frac {{2}^{-n-1} \left ( \arcsin \left ( x \right ) \right ) ^{n} \left ( 2\,n+6 \right ) x}{\sqrt {\pi } \left ( n+1 \right ) \left ( n+3 \right ) }}+{\frac { \left ( \arcsin \left ( x \right ) \right ) ^{n}{2}^{-n}\sqrt {-{x}^{2}+1} \left ( \arcsin \left ( x \right ) {x}^{2}-\arcsin \left ( x \right ) +\sqrt {-{x}^{2}+1}x \right ) }{\sqrt {\pi } \left ( n+1 \right ) \left ( {x}^{2}-1 \right ) }}+{\frac {{2}^{-n}\sqrt {\arcsin \left ( x \right ) }n\LommelS 1 \left ( n+1/2,3/2,\arcsin \left ( x \right ) \right ) x}{\sqrt {\pi } \left ( n+1 \right ) }}-{\frac {{2}^{-n}\sqrt {-{x}^{2}+1} \left ( \arcsin \left ( x \right ) {x}^{2}-\arcsin \left ( x \right ) +\sqrt {-{x}^{2}+1}x \right ) \LommelS 1 \left ( n+3/2,1/2,\arcsin \left ( x \right ) \right ) }{\sqrt {\pi } \left ( n+1 \right ) \sqrt {\arcsin \left ( x \right ) } \left ( {x}^{2}-1 \right ) }} \right ) -2\,ax}} \right ) } \right ) \]
____________________________________________________________________________________
Added January 20, 2019.
Problem 2.7.1.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + \left ( y^2+ \lambda x (\arcsin x)^n y + \lambda ( \arcsin y)^n \right ) w_y = 0 \]
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*Arcsin[x]^n*y + lambda*Arcsin[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*arcsin(x)^n*y + lambda*arcsin(x)^n)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1}{yx+1} \left ( yx\int \!{{\rm e}^{\int \!{\frac { \left ( \arcsin \left ( x \right ) \right ) ^{n}\lambda \,{x}^{2}-2}{x}}\,{\rm d}x}}\,{\rm d}x+{{\rm e}^{\int \!{\frac { \left ( \arcsin \left ( x \right ) \right ) ^{n}\lambda \,{x}^{2}-2}{x}}\,{\rm d}x}}x+\int \!{{\rm e}^{\int \!{\frac { \left ( \arcsin \left ( x \right ) \right ) ^{n}\lambda \,{x}^{2}-2}{x}}\,{\rm d}x}}\,{\rm d}x \right ) } \right ) \]
____________________________________________________________________________________
Added January 20, 2019.
Problem 2.7.1.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x - \left ( (k+1) x^k y^2 - \lambda (\arcsin x)^n (x^{k+1} y-1) \right ) w_y = 0 \]
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*Arcsin[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*arcsin(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'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1}{{x}^{k+1}y-1} \left ( y{x}^{k+1}\int \!{\frac {{{\rm e}^{\lambda \,\int \!{x}^{k+1} \left ( \arcsin \left ( x \right ) \right ) ^{n}\,{\rm d}x}}}{{x}^{k}{x}^{2}}}\,{\rm d}xk+y{x}^{k+1}\int \!{\frac {{{\rm e}^{\lambda \,\int \!{x}^{k+1} \left ( \arcsin \left ( x \right ) \right ) ^{n}\,{\rm d}x}}}{{x}^{k}{x}^{2}}}\,{\rm d}x-{x}^{k+1}{{\rm e}^{\int \!{\frac {{x}^{k+1} \left ( \arcsin \left ( x \right ) \right ) ^{n}\lambda \,x-2\,k-2}{x}}\,{\rm d}x}}-\int \!{\frac {{{\rm e}^{\lambda \,\int \!{x}^{k+1} \left ( \arcsin \left ( x \right ) \right ) ^{n}\,{\rm d}x}}}{{x}^{k}{x}^{2}}}\,{\rm d}xk-\int \!{\frac {{{\rm e}^{\lambda \,\int \!{x}^{k+1} \left ( \arcsin \left ( x \right ) \right ) ^{n}\,{\rm d}x}}}{{x}^{k}{x}^{2}}}\,{\rm d}x \right ) } \right ) \]
____________________________________________________________________________________
Added January 20, 2019.
Problem 2.7.1.8 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + \left ( \lambda (\arcsin x)^n y^2 + a y+ a b -b^2 \lambda (\arcsin x)^n \right ) w_y = 0 \]
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*Arcsin[x]^n*y^2 + a*y + a*b - b^2*lambda*Arcsin[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*arcsin(x)^n*y^2 + a*y+ a*b -b^2 * lambda*arcsin(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 ( y\lambda \,\int \! \left ( \arcsin \left ( x \right ) \right ) ^{n}{{\rm e}^{-2\,\lambda \,b{2}^{n}\sqrt {\pi } \left ( {\frac {{2}^{-n-1} \left ( \arcsin \left ( x \right ) \right ) ^{n} \left ( 2\,n+6 \right ) x}{\sqrt {\pi } \left ( n+1 \right ) \left ( n+3 \right ) }}+{\frac { \left ( \arcsin \left ( x \right ) \right ) ^{n}{2}^{-n}\sqrt {-{x}^{2}+1} \left ( \arcsin \left ( x \right ) {x}^{2}-\arcsin \left ( x \right ) +\sqrt {-{x}^{2}+1}x \right ) }{\sqrt {\pi } \left ( n+1 \right ) \left ( {x}^{2}-1 \right ) }}+{\frac {{2}^{-n}\sqrt {\arcsin \left ( x \right ) }n\LommelS 1 \left ( n+1/2,3/2,\arcsin \left ( x \right ) \right ) x}{\sqrt {\pi } \left ( n+1 \right ) }}-{\frac {{2}^{-n}\sqrt {-{x}^{2}+1} \left ( \arcsin \left ( x \right ) {x}^{2}-\arcsin \left ( x \right ) +\sqrt {-{x}^{2}+1}x \right ) \LommelS 1 \left ( n+3/2,1/2,\arcsin \left ( x \right ) \right ) }{\sqrt {\pi } \left ( n+1 \right ) \sqrt {\arcsin \left ( x \right ) } \left ( {x}^{2}-1 \right ) }} \right ) +ax}}\,{\rm d}x+\lambda \,\int \! \left ( \arcsin \left ( x \right ) \right ) ^{n}{{\rm e}^{-2\,\lambda \,b{2}^{n}\sqrt {\pi } \left ( {\frac {{2}^{-n-1} \left ( \arcsin \left ( x \right ) \right ) ^{n} \left ( 2\,n+6 \right ) x}{\sqrt {\pi } \left ( n+1 \right ) \left ( n+3 \right ) }}+{\frac { \left ( \arcsin \left ( x \right ) \right ) ^{n}{2}^{-n}\sqrt {-{x}^{2}+1} \left ( \arcsin \left ( x \right ) {x}^{2}-\arcsin \left ( x \right ) +\sqrt {-{x}^{2}+1}x \right ) }{\sqrt {\pi } \left ( n+1 \right ) \left ( {x}^{2}-1 \right ) }}+{\frac {{2}^{-n}\sqrt {\arcsin \left ( x \right ) }n\LommelS 1 \left ( n+1/2,3/2,\arcsin \left ( x \right ) \right ) x}{\sqrt {\pi } \left ( n+1 \right ) }}-{\frac {{2}^{-n}\sqrt {-{x}^{2}+1} \left ( \arcsin \left ( x \right ) {x}^{2}-\arcsin \left ( x \right ) +\sqrt {-{x}^{2}+1}x \right ) \LommelS 1 \left ( n+3/2,1/2,\arcsin \left ( x \right ) \right ) }{\sqrt {\pi } \left ( n+1 \right ) \sqrt {\arcsin \left ( x \right ) } \left ( {x}^{2}-1 \right ) }} \right ) +ax}}\,{\rm d}xb+{{\rm e}^{-2\,\lambda \,b{2}^{n}\sqrt {\pi } \left ( {\frac {{2}^{-n-1} \left ( \arcsin \left ( x \right ) \right ) ^{n} \left ( 2\,n+6 \right ) x}{\sqrt {\pi } \left ( n+1 \right ) \left ( n+3 \right ) }}+{\frac { \left ( \arcsin \left ( x \right ) \right ) ^{n}{2}^{-n}\sqrt {-{x}^{2}+1} \left ( \arcsin \left ( x \right ) {x}^{2}-\arcsin \left ( x \right ) +\sqrt {-{x}^{2}+1}x \right ) }{\sqrt {\pi } \left ( n+1 \right ) \left ( {x}^{2}-1 \right ) }}+{\frac {{2}^{-n}\sqrt {\arcsin \left ( x \right ) }n\LommelS 1 \left ( n+1/2,3/2,\arcsin \left ( x \right ) \right ) x}{\sqrt {\pi } \left ( n+1 \right ) }}-{\frac {{2}^{-n}\sqrt {-{x}^{2}+1} \left ( \arcsin \left ( x \right ) {x}^{2}-\arcsin \left ( x \right ) +\sqrt {-{x}^{2}+1}x \right ) \LommelS 1 \left ( n+3/2,1/2,\arcsin \left ( x \right ) \right ) }{\sqrt {\pi } \left ( n+1 \right ) \sqrt {\arcsin \left ( x \right ) } \left ( {x}^{2}-1 \right ) }} \right ) +ax}} \right ) } \right ) \]
____________________________________________________________________________________
Added January 29, 2019.
Problem 2.7.1.9 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + \left ( \lambda (\arcsin x)^n y^2 - b \lambda x^m (\arcsin x)^n y+ b m x^{m-1} \right ) w_y = 0 \]
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*Arcsin[x]^n*y^2 - b*lambda*x^m*ArcSin[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*arcsin(x)^n*y^2 - b*lambda*x^m*arcsin(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 2.7.1.10 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + \left ( \lambda (\arcsin x)^n y^2 + b m x^{m-1} - \lambda b^2 x^{2 m} (\arcsin x)^n \right ) w_y = 0 \]
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*ArcSin[x]^n*y^2 + b*m*x^(m - 1) - lambda*b^2*x^(2*m)*ArcSin[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*arcsin(x)^n*y^2 + b*m*x^(m-1) - lambda*b^2*x^(2*m)*arcsin(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 2.7.1.11 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + \left ( \lambda (\arcsin x)^n (y - a x^m -b)^2 + a m x^{m-1} \right ) w_y = 0 \]
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*ArcSin[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]];
\[ \text {\$Aborted} \]
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*arcsin(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 {{2}^{n}{2}^{-n} \left ( \arcsin \left ( x \right ) \right ) ^{3/2}\sqrt {-{x}^{2}+1}{x}^{m} \left ( \arcsin \left ( x \right ) \right ) ^{n}a\lambda +\LommelS 1 \left ( n+1/2,3/2,\arcsin \left ( x \right ) \right ) {2}^{n}{2}^{-n}\arcsin \left ( x \right ) {x}^{m}a\lambda \,nx+{2}^{n}{2}^{-n} \left ( \arcsin \left ( x \right ) \right ) ^{3/2}\sqrt {-{x}^{2}+1} \left ( \arcsin \left ( x \right ) \right ) ^{n}b\lambda -{2}^{n}{2}^{-n} \left ( \arcsin \left ( x \right ) \right ) ^{3/2}\sqrt {-{x}^{2}+1} \left ( \arcsin \left ( x \right ) \right ) ^{n}\lambda \,y+\LommelS 1 \left ( n+1/2,3/2,\arcsin \left ( x \right ) \right ) {2}^{n}{2}^{-n}\arcsin \left ( x \right ) b\lambda \,nx-\LommelS 1 \left ( n+1/2,3/2,\arcsin \left ( x \right ) \right ) {2}^{n}{2}^{-n}\arcsin \left ( x \right ) \lambda \,nxy-{2}^{n}\LommelS 1 \left ( n+3/2,1/2,\arcsin \left ( x \right ) \right ) {2}^{-n}\sqrt {-{x}^{2}+1}\arcsin \left ( x \right ) {x}^{m}a\lambda +2\,{2}^{n}{2}^{-n-1}\sqrt {\arcsin \left ( x \right ) }{x}^{m} \left ( \arcsin \left ( x \right ) \right ) ^{n}a\lambda \,x-{2}^{n}{2}^{-n}\sqrt {\arcsin \left ( x \right ) }{x}^{m} \left ( \arcsin \left ( x \right ) \right ) ^{n}a\lambda \,x-{2}^{n}\LommelS 1 \left ( n+3/2,1/2,\arcsin \left ( x \right ) \right ) {2}^{-n}\sqrt {-{x}^{2}+1}\arcsin \left ( x \right ) b\lambda +{2}^{n}\LommelS 1 \left ( n+3/2,1/2,\arcsin \left ( x \right ) \right ) {2}^{-n}\sqrt {-{x}^{2}+1}\arcsin \left ( x \right ) \lambda \,y+{2}^{n}\LommelS 1 \left ( n+3/2,1/2,\arcsin \left ( x \right ) \right ) {2}^{-n}{x}^{m}a\lambda \,x+2\,{2}^{n}{2}^{-n-1}\sqrt {\arcsin \left ( x \right ) } \left ( \arcsin \left ( x \right ) \right ) ^{n}b\lambda \,x-2\,{2}^{n}{2}^{-n-1}\sqrt {\arcsin \left ( x \right ) } \left ( \arcsin \left ( x \right ) \right ) ^{n}\lambda \,xy-{2}^{n}{2}^{-n}\sqrt {\arcsin \left ( x \right ) } \left ( \arcsin \left ( x \right ) \right ) ^{n}b\lambda \,x+{2}^{n}{2}^{-n}\sqrt {\arcsin \left ( x \right ) } \left ( \arcsin \left ( x \right ) \right ) ^{n}\lambda \,xy+{2}^{n}\LommelS 1 \left ( n+3/2,1/2,\arcsin \left ( x \right ) \right ) {2}^{-n}b\lambda \,x-{2}^{n}\LommelS 1 \left ( n+3/2,1/2,\arcsin \left ( x \right ) \right ) {2}^{-n}\lambda \,xy-\sqrt {\arcsin \left ( x \right ) }n-\sqrt {\arcsin \left ( x \right ) }}{ \left ( n+1 \right ) \sqrt {\arcsin \left ( x \right ) } \left ( a{x}^{m}+b-y \right ) }} \right ) \]
____________________________________________________________________________________
Added January 29, 2019.
Problem 2.7.1.12 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ x w_x + \left ( \lambda (\arcsin x)^n y^2 +k y+ \lambda b^2 x^{2 k} (\arcsin x)^n \right ) w_y = 0 \]
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*ArcSin[x]^n*y^2 + k*y + lambda*b^2*x^(2*k)*ArcSin[x]^n)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to c_1\left (\tan ^{-1}\left (\frac {y x^{-k}}{\sqrt {b^2}}\right )-\sqrt {b^2} \int _1^x \lambda K[1]^{k-1} \sin ^{-1}(K[1])^n \, dK[1]\right )\right \}\right \} \]
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*arcsin(x)^n*y^2 +k*y+ lambda*b^2*x^(2*k)*arcsin(x)^n )*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( \lambda \,b\int \! \left ( \arcsin \left ( x \right ) \right ) ^{n}{x}^{k-1}\,{\rm d}x-\arctan \left ( {\frac {{x}^{-k}y}{b}} \right ) \right ) \]