Added April 13, 2019.
Problem Chapter 5.7.1.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b w_y = w + c_1 \arcsin ^k(\lambda x) + c_2 \arcsin ^n(\beta y) \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*D[w[x, y], y] == w[x,y]+ c1*ArcSin[lambda*x]^k+c2*ArcSin[beta*y]^n; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to e^{\frac {x}{a}} \left (\int _1^x\frac {e^{-\frac {K[1]}{a}} \left (\text {c1} \sin ^{-1}(\lambda K[1])^k+\text {c2} \sin ^{-1}\left (\beta \left (y+\frac {b (K[1]-x)}{a}\right )\right )^n\right )}{a}dK[1]+c_1\left (y-\frac {b x}{a}\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x)+ b*diff(w(x,y),y) = w(x,y)+c1*arcsin(lambda*x)^k+c2*arcsin(beta*y)^n; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac {1}{a}{{\rm e}^{-{\frac {{\it \_a}}{a}}}} \left ( {\it c1}\, \left ( \arcsin \left ( {\it \_a}\,\lambda \right ) \right ) ^{k}+{\it c2}\, \left ( \arcsin \left ( {\frac {\beta \, \left ( ay-b \left ( x-{\it \_a} \right ) \right ) }{a}} \right ) \right ) ^{n} \right ) }{d{\it \_a}}+{\it \_F1} \left ( {\frac {ay-xb}{a}} \right ) \right ) {{\rm e}^{{\frac {x}{a}}}}\]
____________________________________________________________________________________
Added April 13, 2019.
Problem Chapter 5.7.1.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b w_y = c w + \arcsin ^k(\lambda x) \arcsin ^n(\beta y) \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*D[w[x, y], y] == c*w[x,y]+ ArcSin[lambda*x]^k*ArcSin[beta*y]^n; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to e^{\frac {c x}{a}} \left (\int _1^x\frac {e^{-\frac {c K[1]}{a}} \sin ^{-1}(\lambda K[1])^k \sin ^{-1}\left (\beta \left (y+\frac {b (K[1]-x)}{a}\right )\right )^n}{a}dK[1]+c_1\left (y-\frac {b x}{a}\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x)+ b*diff(w(x,y),y) = c*w(x,y)+ arcsin(lambda*x)^k*arcsin(beta*y)^n; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac { \left ( \arcsin \left ( {\it \_a}\,\lambda \right ) \right ) ^{k}}{a} \left ( \arcsin \left ( {\frac {\beta \, \left ( ay-b \left ( x-{\it \_a} \right ) \right ) }{a}} \right ) \right ) ^{n}{{\rm e}^{-{\frac {c{\it \_a}}{a}}}}}{d{\it \_a}}+{\it \_F1} \left ( {\frac {ay-xb}{a}} \right ) \right ) {{\rm e}^{{\frac {cx}{a}}}}\]
____________________________________________________________________________________
Added April 13, 2019.
Problem Chapter 5.7.1.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b w_y = \left ( c_1 \arcsin (\lambda _1 x) + c_2 \arcsin (\lambda _2 y)\right ) w+ s_1 \arcsin ^n(\beta _1 x)+ s_2 \arcsin ^k(\beta _2 y) \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*D[w[x, y], y] == ( c1*ArcSin[lambda1*x] + c2*ArcSin[lambda2*y])*w[x,y]+ s1*ArcSin[beta1*x]^n+ s2*ArcSin[beta2*y]^k; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \exp \left (\frac {\text {c1} \sqrt {1-\text {lambda1}^2 x^2}}{a \text {lambda1}}+\frac {\text {c1} x \sin ^{-1}(\text {lambda1} x)}{a}+\frac {\text {c2} \sqrt {1-\text {lambda2}^2 y^2}}{b \text {lambda2}}+\frac {\text {c2} y \sin ^{-1}(\text {lambda2} y)}{b}\right ) \left (\int _1^x\frac {\exp \left (-\frac {b \text {c1} \text {lambda2} \sin ^{-1}(\text {lambda1} K[1]) K[1] \text {lambda1}+\text {c2} \text {lambda2} \sin ^{-1}\left (\text {lambda2} \left (y+\frac {b (K[1]-x)}{a}\right )\right ) (a y+b (K[1]-x)) \text {lambda1}+a \text {c2} \sqrt {-y^2 \text {lambda2}^2-\frac {b^2 (x-K[1])^2 \text {lambda2}^2}{a^2}+\frac {2 b y (x-K[1]) \text {lambda2}^2}{a}+1} \text {lambda1}+b \text {c1} \text {lambda2} \sqrt {1-\text {lambda1}^2 K[1]^2}}{a b \text {lambda1} \text {lambda2}}\right ) \left (\text {s2} \sin ^{-1}\left (\text {beta2} \left (y+\frac {b (K[1]-x)}{a}\right )\right )^k+\text {s1} \sin ^{-1}(\text {beta1} K[1])^n\right )}{a}dK[1]+c_1\left (y-\frac {b x}{a}\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x)+ b*diff(w(x,y),y) = ( c1*arcsin(lambda1*x) + c2*arcsin(lambda2*y))*w(x,y)+ s1*arcsin(beta1*x)^n+ s2*arcsin(beta2*y)^k; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac {1}{a}{{\rm e}^{{\frac {1}{a\lambda 1\,b\lambda 2} \left ( -\sqrt {-{\frac { \left ( \left ( \lambda 2\,y-1 \right ) a-b\lambda 2\, \left ( x-{\it \_a} \right ) \right ) \left ( \left ( \lambda 2\,y+1 \right ) a-b\lambda 2\, \left ( x-{\it \_a} \right ) \right ) }{{a}^{2}}}}a{\it c2}\,\lambda 1-\lambda 2\, \left ( \lambda 1\, \left ( \left ( -x+{\it \_a} \right ) b+ay \right ) {\it c2}\,\arcsin \left ( {\frac {\lambda 2\, \left ( ay-b \left ( x-{\it \_a} \right ) \right ) }{a}} \right ) +b{\it c1}\, \left ( \arcsin \left ( \lambda 1\,{\it \_a} \right ) {\it \_a}\,\lambda 1+\sqrt {-{{\it \_a}}^{2}{\lambda 1}^{2}+1} \right ) \right ) \right ) }}} \left ( {\it s1}\, \left ( \arcsin \left ( \beta 1\,{\it \_a} \right ) \right ) ^{n}+{\it s2}\, \left ( \arcsin \left ( {\frac {\beta 2\, \left ( ay-b \left ( x-{\it \_a} \right ) \right ) }{a}} \right ) \right ) ^{k} \right ) }{d{\it \_a}}+{\it \_F1} \left ( {\frac {ay-xb}{a}} \right ) \right ) {{\rm e}^{{\frac {1}{a\lambda 1\,b\lambda 2} \left ( \sqrt {-{\lambda 2}^{2}{y}^{2}+1}a{\it c2}\,\lambda 1+\lambda 2\, \left ( b{\it c1}\,\sqrt {-{\lambda 1}^{2}{x}^{2}+1}+\lambda 1\, \left ( a\arcsin \left ( \lambda 2\,y \right ) y{\it c2}+bx{\it c1}\,\arcsin \left ( \lambda 1\,x \right ) \right ) \right ) \right ) }}}\]
____________________________________________________________________________________
Added April 13, 2019.
Problem Chapter 5.7.1.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b \arcsin ^m(\mu x) w_y = c \arcsin ^k(\nu x) w + p \arcsin ^n(\beta y) \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*ArcSin[mu*x]^m*D[w[x, y], y] == c*ArcSin[nu*x]^k*w[x,y]+p*ArcSin[beta*y]^n; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \exp \left (\frac {i c \sin ^{-1}(\nu x)^k \left (\sin ^{-1}(\nu x)^2\right )^{-k} \left (\left (-i \sin ^{-1}(\nu x)\right )^k \text {Gamma}\left (k+1,i \sin ^{-1}(\nu x)\right )-\left (i \sin ^{-1}(\nu x)\right )^k \text {Gamma}\left (k+1,-i \sin ^{-1}(\nu x)\right )\right )}{2 a \nu }\right ) \left (c_1\left (y-\int _1^x\frac {b \sin ^{-1}(\mu K[1])^m}{a}dK[1]\right )+\int _1^x\frac {\exp \left (\frac {i c \sin ^{-1}(\nu K[2])^k \left (\sin ^{-1}(\nu K[2])^2\right )^{-k} \left (\left (i \sin ^{-1}(\nu K[2])\right )^k \text {Gamma}\left (k+1,-i \sin ^{-1}(\nu K[2])\right )-\left (-i \sin ^{-1}(\nu K[2])\right )^k \text {Gamma}\left (k+1,i \sin ^{-1}(\nu K[2])\right )\right )}{2 a \nu }\right ) p \sin ^{-1}\left (\beta \left (y-\int _1^x\frac {b \sin ^{-1}(\mu K[1])^m}{a}dK[1]+\int _1^{K[2]}\frac {b \sin ^{-1}(\mu K[1])^m}{a}dK[1]\right )\right ){}^n}{a}dK[2]\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x)+ b*arcsin(mu*x)^m*diff(w(x,y),y) = c*arcsin(nu*x)^k*w(x,y)+p*arcsin(beta*y)^n; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac {p}{a} \left ( -\arcsin \left ( {\frac { \left ( \mu \,{\it \_f}+1 \right ) \beta \, \left ( \mu \,{\it \_f}-1 \right ) }{ \left ( m+1 \right ) \mu \, \left ( {{\it \_f}}^{2}{\mu }^{2}-1 \right ) a} \left ( -{2}^{-m}\arcsin \left ( \mu \,{\it \_f} \right ) b{2}^{m} \left ( -{\LommelS 1 \left ( m+{\frac {3}{2}},{\frac {1}{2}},\arcsin \left ( \mu \,{\it \_f} \right ) \right ) {\frac {1}{\sqrt {\arcsin \left ( \mu \,{\it \_f} \right ) }}}}+ \left ( \arcsin \left ( \mu \,{\it \_f} \right ) \right ) ^{m} \right ) \sqrt {-{{\it \_f}}^{2}{\mu }^{2}+1}+ \left ( a \left ( m+1 \right ) \int \!{\frac {b \left ( \arcsin \left ( \mu \,x \right ) \right ) ^{m}}{a}}\,{\rm d}x-{{2}^{-m}b{\it \_f}\,{2}^{m}\LommelS 1 \left ( m+{\frac {3}{2}},{\frac {1}{2}},\arcsin \left ( \mu \,{\it \_f} \right ) \right ) {\frac {1}{\sqrt {\arcsin \left ( \mu \,{\it \_f} \right ) }}}}-{2}^{m}{2}^{-m}b\sqrt {\arcsin \left ( \mu \,{\it \_f} \right ) }\LommelS 1 \left ( m+{\frac {1}{2}},{\frac {3}{2}},\arcsin \left ( \mu \,{\it \_f} \right ) \right ) m{\it \_f}-a \left ( m+1 \right ) y \right ) \mu \right ) } \right ) \right ) ^{n}{{\rm e}^{{\frac {{2}^{k}c \left ( \nu \,{\it \_f}+1 \right ) \left ( \nu \,{\it \_f}-1 \right ) {2}^{-k}}{ \left ( 1+k \right ) a\nu \, \left ( {{\it \_f}}^{2}{\nu }^{2}-1 \right ) } \left ( \arcsin \left ( \nu \,{\it \_f} \right ) \left ( {\LommelS 1 \left ( k+{\frac {3}{2}},{\frac {1}{2}},\arcsin \left ( \nu \,{\it \_f} \right ) \right ) {\frac {1}{\sqrt {\arcsin \left ( \nu \,{\it \_f} \right ) }}}}- \left ( \arcsin \left ( \nu \,{\it \_f} \right ) \right ) ^{k} \right ) \sqrt {-{{\it \_f}}^{2}{\nu }^{2}+1}-\nu \,{\it \_f}\, \left ( \sqrt {\arcsin \left ( \nu \,{\it \_f} \right ) }k\LommelS 1 \left ( k+{\frac {1}{2}},{\frac {3}{2}},\arcsin \left ( \nu \,{\it \_f} \right ) \right ) +{\LommelS 1 \left ( k+{\frac {3}{2}},{\frac {1}{2}},\arcsin \left ( \nu \,{\it \_f} \right ) \right ) {\frac {1}{\sqrt {\arcsin \left ( \nu \,{\it \_f} \right ) }}}} \right ) \right ) }}}}{d{\it \_f}}+{\it \_F1} \left ( {\frac {1}{ \left ( m+1 \right ) a\mu } \left ( -b \left ( -\arcsin \left ( \mu \,x \right ) \LommelS 1 \left ( m+{\frac {3}{2}},{\frac {1}{2}},\arcsin \left ( \mu \,x \right ) \right ) + \left ( \arcsin \left ( \mu \,x \right ) \right ) ^{m+{\frac {3}{2}}} \right ) \sqrt {-{\mu }^{2}{x}^{2}+1}+\mu \, \left ( -bx\LommelS 1 \left ( m+{\frac {3}{2}},{\frac {1}{2}},\arcsin \left ( \mu \,x \right ) \right ) -\LommelS 1 \left ( m+{\frac {1}{2}},{\frac {3}{2}},\arcsin \left ( \mu \,x \right ) \right ) bmx\arcsin \left ( \mu \,x \right ) +a\sqrt {\arcsin \left ( \mu \,x \right ) }y \left ( m+1 \right ) \right ) \right ) {\frac {1}{\sqrt {\arcsin \left ( \mu \,x \right ) }}}} \right ) \right ) {{\rm e}^{\int \!{\frac { \left ( \arcsin \left ( \nu \,x \right ) \right ) ^{k}c}{a}}\,{\rm d}x}}\]
____________________________________________________________________________________
Added April 13, 2019.
Problem Chapter 5.7.1.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b \arcsin ^m(\mu x) w_y = c \arcsin ^k(\nu y) w + p \arcsin ^n(\beta x) \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*ArcSin[mu*x]^m*D[w[x, y], y] == c*ArcSin[nu*y]^k*w[x,y]+p*ArcSin[beta*x]^n; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \exp \left (\int _1^x\frac {c \sin ^{-1}\left (\nu \left (y-\int _1^x\frac {b \sin ^{-1}(\mu K[1])^m}{a}dK[1]+\int _1^{K[2]}\frac {b \sin ^{-1}(\mu K[1])^m}{a}dK[1]\right )\right ){}^k}{a}dK[2]\right ) \left (\int _1^x\frac {\exp \left (-\int _1^{K[3]}\frac {c \sin ^{-1}\left (\nu \left (y-\int _1^x\frac {b \sin ^{-1}(\mu K[1])^m}{a}dK[1]+\int _1^{K[2]}\frac {b \sin ^{-1}(\mu K[1])^m}{a}dK[1]\right )\right ){}^k}{a}dK[2]\right ) p \sin ^{-1}(\beta K[3])^n}{a}dK[3]+c_1\left (y-\int _1^x\frac {b \sin ^{-1}(\mu K[1])^m}{a}dK[1]\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x)+ b*arcsin(mu*x)^m*diff(w(x,y),y) = c*arcsin(nu*y)^k*w(x,y)+p*arcsin(beta*x)^n; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac {p \left ( \arcsin \left ( \beta \,{\it \_f} \right ) \right ) ^{n}}{a}{{\rm e}^{-{\frac {c}{a}\int \! \left ( -\arcsin \left ( {\frac { \left ( \mu \,{\it \_f}+1 \right ) \nu \, \left ( \mu \,{\it \_f}-1 \right ) }{ \left ( m+1 \right ) \mu \, \left ( {{\it \_f}}^{2}{\mu }^{2}-1 \right ) a} \left ( -{2}^{-m}\arcsin \left ( \mu \,{\it \_f} \right ) b{2}^{m} \left ( -{\LommelS 1 \left ( m+{\frac {3}{2}},{\frac {1}{2}},\arcsin \left ( \mu \,{\it \_f} \right ) \right ) {\frac {1}{\sqrt {\arcsin \left ( \mu \,{\it \_f} \right ) }}}}+ \left ( \arcsin \left ( \mu \,{\it \_f} \right ) \right ) ^{m} \right ) \sqrt {-{{\it \_f}}^{2}{\mu }^{2}+1}+ \left ( a \left ( m+1 \right ) \int \!{\frac {b \left ( \arcsin \left ( \mu \,x \right ) \right ) ^{m}}{a}}\,{\rm d}x-{{2}^{-m}b{\it \_f}\,{2}^{m}\LommelS 1 \left ( m+{\frac {3}{2}},{\frac {1}{2}},\arcsin \left ( \mu \,{\it \_f} \right ) \right ) {\frac {1}{\sqrt {\arcsin \left ( \mu \,{\it \_f} \right ) }}}}-{2}^{m}{2}^{-m}b\sqrt {\arcsin \left ( \mu \,{\it \_f} \right ) }\LommelS 1 \left ( m+{\frac {1}{2}},{\frac {3}{2}},\arcsin \left ( \mu \,{\it \_f} \right ) \right ) m{\it \_f}-a \left ( m+1 \right ) y \right ) \mu \right ) } \right ) \right ) ^{k}\,{\rm d}{\it \_f}}}}}{d{\it \_f}}+{\it \_F1} \left ( {\frac {1}{ \left ( m+1 \right ) a\mu } \left ( -b \left ( -\arcsin \left ( \mu \,x \right ) \LommelS 1 \left ( m+{\frac {3}{2}},{\frac {1}{2}},\arcsin \left ( \mu \,x \right ) \right ) + \left ( \arcsin \left ( \mu \,x \right ) \right ) ^{m+{\frac {3}{2}}} \right ) \sqrt {-{\mu }^{2}{x}^{2}+1}+\mu \, \left ( -bx\LommelS 1 \left ( m+{\frac {3}{2}},{\frac {1}{2}},\arcsin \left ( \mu \,x \right ) \right ) -\LommelS 1 \left ( m+{\frac {1}{2}},{\frac {3}{2}},\arcsin \left ( \mu \,x \right ) \right ) bmx\arcsin \left ( \mu \,x \right ) +a\sqrt {\arcsin \left ( \mu \,x \right ) }y \left ( m+1 \right ) \right ) \right ) {\frac {1}{\sqrt {\arcsin \left ( \mu \,x \right ) }}}} \right ) \right ) {{\rm e}^{\int ^{x}\!{\frac {c}{a} \left ( -\arcsin \left ( {\frac { \left ( \mu \,{\it \_b}+1 \right ) \nu \, \left ( \mu \,{\it \_b}-1 \right ) }{ \left ( m+1 \right ) \mu \, \left ( {{\it \_b}}^{2}{\mu }^{2}-1 \right ) a} \left ( -{2}^{-m}\arcsin \left ( \mu \,{\it \_b} \right ) b{2}^{m} \left ( -{\LommelS 1 \left ( m+{\frac {3}{2}},{\frac {1}{2}},\arcsin \left ( \mu \,{\it \_b} \right ) \right ) {\frac {1}{\sqrt {\arcsin \left ( \mu \,{\it \_b} \right ) }}}}+ \left ( \arcsin \left ( \mu \,{\it \_b} \right ) \right ) ^{m} \right ) \sqrt {-{{\it \_b}}^{2}{\mu }^{2}+1}+ \left ( a \left ( m+1 \right ) \int \!{\frac {b \left ( \arcsin \left ( \mu \,x \right ) \right ) ^{m}}{a}}\,{\rm d}x-{{2}^{-m}b{\it \_b}\,{2}^{m}\LommelS 1 \left ( m+{\frac {3}{2}},{\frac {1}{2}},\arcsin \left ( \mu \,{\it \_b} \right ) \right ) {\frac {1}{\sqrt {\arcsin \left ( \mu \,{\it \_b} \right ) }}}}-{2}^{m}{2}^{-m}b\sqrt {\arcsin \left ( \mu \,{\it \_b} \right ) }\LommelS 1 \left ( m+{\frac {1}{2}},{\frac {3}{2}},\arcsin \left ( \mu \,{\it \_b} \right ) \right ) m{\it \_b}-a \left ( m+1 \right ) y \right ) \mu \right ) } \right ) \right ) ^{k}}{d{\it \_b}}}}\]
____________________________________________________________________________________