Added March 9, 2019.
Problem Chapter 4.7.1.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b w_y = \left ( c \arcsin (\frac {x}{\lambda } + k \arcsin (\frac {y}{\beta } ) \right ) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*D[w[x, y], y] == (c*ArcSin[x/lambda] + k*ArcSin[y/beta])*w[x, y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (y-\frac {b x}{a}\right ) \exp \left (\frac {\frac {k \left (\sqrt {a^2 \left (\beta ^2-y^2\right )} (a y-b x) \tan ^{-1}\left (\frac {a y}{\sqrt {a^2 \left (\beta ^2-y^2\right )}}\right )+a^2 \left (\beta ^2-y^2\right )\right )}{b \beta \sqrt {1-\frac {y^2}{\beta ^2}}}+a k x \sin ^{-1}\left (\frac {y}{\beta }\right )+a c \lambda \sqrt {1-\frac {x^2}{\lambda ^2}}+a c x \sin ^{-1}\left (\frac {x}{\lambda }\right )}{a^2}\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x)+ b*diff(w(x,y),y) = (c*arcsin(x/lambda)+k*arcsin(y/beta))*w(x,y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {ay-xb}{a}} \right ) {{\rm e}^{{\frac {1}{ba} \left ( a\arcsin \left ( {\frac {y}{\beta }} \right ) ky+\sqrt {{\frac {{\beta }^{2}-{y}^{2}}{{\beta }^{2}}}}a\beta \,k+\sqrt {-{\frac {{x}^{2}}{{\lambda }^{2}}}+1}bc\lambda +\arcsin \left ( {\frac {x}{\lambda }} \right ) bcx \right ) }}}\]
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Added March 9, 2019.
Problem Chapter 4.7.1.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b w_y = c \arcsin (\lambda x+\beta y) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*D[w[x, y], y] == c*ArcSin[lambda*x + beta*y]*w[x, y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (y-\frac {b x}{a}\right ) \exp \left (\frac {c \left (\sqrt {-\beta ^2 y^2-2 \beta \lambda x y-\lambda ^2 x^2+1}+(\beta y+\lambda x) \sin ^{-1}(\beta y+\lambda x)\right )}{a \lambda +b \beta }\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x)+ b*diff(w(x,y),y) = c*arcsin(lambda*x+beta*y)*w(x,y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {ay-xb}{a}} \right ) {{\rm e}^{{\frac {c}{a\lambda +\beta \,b} \left ( \sqrt {-{\beta }^{2}{y}^{2}-2\,\beta \,\lambda \,xy-{x}^{2}{\lambda }^{2}+1}+\arcsin \left ( \beta \,y+x\lambda \right ) \left ( \beta \,y+x\lambda \right ) \right ) }}}\]
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Added March 9, 2019.
Problem Chapter 4.7.1.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b w_y = a x \arcsin (\lambda x+\beta y) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*D[w[x, y], y] == a*x*ArcSin[lambda*x + beta*y]*w[x, y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (y-\frac {b x}{a}\right ) \exp \left (\frac {a \left (\sqrt {-\beta ^2 y^2-2 \beta \lambda x y-\lambda ^2 x^2+1} (-3 a \beta y+a \lambda x+4 b \beta x)+\sin ^{-1}(\beta y+\lambda x) \left (a \left (-2 \beta ^2 y^2+2 \lambda ^2 x^2-1\right )+4 b \beta x (\beta y+\lambda x)\right )\right )}{4 (a \lambda +b \beta )^2}\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x)+ b*diff(w(x,y),y) = a*x*arcsin(lambda*x+beta*y)*w(x,y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {ay-xb}{a}} \right ) {{\rm e}^{{\frac {a}{2\, \left ( a\lambda +\beta \,b \right ) ^{2}} \left ( \left ( \left ( {\frac {x\lambda }{2}}-{\frac {3\,\beta \,y}{2}} \right ) a+2\,b\beta \,x \right ) \sqrt {-{\beta }^{2}{y}^{2}-2\,\beta \,\lambda \,xy-{x}^{2}{\lambda }^{2}+1}+ \left ( \left ( {x}^{2}{\lambda }^{2}-{\beta }^{2}{y}^{2}-{\frac {1}{2}} \right ) a+2\,b\beta \,x \left ( \beta \,y+x\lambda \right ) \right ) \arcsin \left ( \beta \,y+x\lambda \right ) \right ) }}}\]
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Added March 9, 2019.
Problem Chapter 4.7.1.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b \arcsin ^n(\lambda x)w_y = \left ( c \arcsin ^m(\mu x) + s \arcsin ^k(\beta y) \right ) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*ArcSin[lambda*x]^n*D[w[x, y], y] == (c*ArcSin[mu*x]^m + s*ArcSin[beta*y]^k)*w[x, y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (y-\int _1^x\frac {b \sin ^{-1}(\lambda K[1])^n}{a}dK[1]\right ) \exp \left (\int _1^x\frac {s \sin ^{-1}\left (\beta \left (y-\int _1^x\frac {b \sin ^{-1}(\lambda K[1])^n}{a}dK[1]+\int _1^{K[2]}\frac {b \sin ^{-1}(\lambda K[1])^n}{a}dK[1]\right )\right ){}^k+c \sin ^{-1}(\mu K[2])^m}{a}dK[2]\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x)+ b*arcsin(lambda*x)^n*diff(w(x,y),y) =(c*arcsin(mu*x)^m+s*arcsin(beta*y)^k)*w(x,y); 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 ( 1+n \right ) a\lambda } \left ( -b \left ( -\arcsin \left ( x\lambda \right ) \LommelS 1 \left ( n+{\frac {3}{2}},{\frac {1}{2}},\arcsin \left ( x\lambda \right ) \right ) + \left ( \arcsin \left ( x\lambda \right ) \right ) ^{n+{\frac {3}{2}}} \right ) \sqrt {-{x}^{2}{\lambda }^{2}+1}+\lambda \, \left ( -bx\LommelS 1 \left ( n+{\frac {3}{2}},{\frac {1}{2}},\arcsin \left ( x\lambda \right ) \right ) -\LommelS 1 \left ( n+{\frac {1}{2}},{\frac {3}{2}},\arcsin \left ( x\lambda \right ) \right ) bnx\arcsin \left ( x\lambda \right ) +a\sqrt {\arcsin \left ( x\lambda \right ) }y \left ( 1+n \right ) \right ) \right ) {\frac {1}{\sqrt {\arcsin \left ( x\lambda \right ) }}}} \right ) {{\rm e}^{\int ^{x}\!{\frac {1}{a} \left ( c \left ( \arcsin \left ( \mu \,{\it \_b} \right ) \right ) ^{m}+s \left ( -\arcsin \left ( {\frac { \left ( {\it \_b}\,\lambda +1 \right ) \left ( {\it \_b}\,\lambda -1 \right ) \beta }{ \left ( 1+n \right ) a\lambda \, \left ( {{\it \_b}}^{2}{\lambda }^{2}-1 \right ) } \left ( -{2}^{-n}\arcsin \left ( {\it \_b}\,\lambda \right ) b{2}^{n} \left ( -{\LommelS 1 \left ( n+{\frac {3}{2}},{\frac {1}{2}},\arcsin \left ( {\it \_b}\,\lambda \right ) \right ) {\frac {1}{\sqrt {\arcsin \left ( {\it \_b}\,\lambda \right ) }}}}+ \left ( \arcsin \left ( {\it \_b}\,\lambda \right ) \right ) ^{n} \right ) \sqrt {-{{\it \_b}}^{2}{\lambda }^{2}+1}+ \left ( a \left ( 1+n \right ) \int \!{\frac {b \left ( \arcsin \left ( x\lambda \right ) \right ) ^{n}}{a}}\,{\rm d}x-{{2}^{-n}b{\it \_b}\,{2}^{n}\LommelS 1 \left ( n+{\frac {3}{2}},{\frac {1}{2}},\arcsin \left ( {\it \_b}\,\lambda \right ) \right ) {\frac {1}{\sqrt {\arcsin \left ( {\it \_b}\,\lambda \right ) }}}}-{2}^{-n}b\sqrt {\arcsin \left ( {\it \_b}\,\lambda \right ) }\LommelS 1 \left ( n+{\frac {1}{2}},{\frac {3}{2}},\arcsin \left ( {\it \_b}\,\lambda \right ) \right ) n{\it \_b}\,{2}^{n}-ay \left ( 1+n \right ) \right ) \lambda \right ) } \right ) \right ) ^{k} \right ) }{d{\it \_b}}}}\]
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Added March 9, 2019.
Problem Chapter 4.7.1.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b \arcsin ^n(\lambda y)w_y = \left ( c \arcsin ^m(\mu x) + s \arcsin ^k(\beta y) \right ) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*ArcSin[lambda*y]^n*D[w[x, y], y] == (c*ArcSin[mu*x]^m + s*ArcSin[beta*y]^k)*w[x, y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (-\frac {b x}{a}-\frac {i \sin ^{-1}(\lambda y)^{-n} \left (\left (-i \sin ^{-1}(\lambda y)\right )^n \text {Gamma}\left (1-n,-i \sin ^{-1}(\lambda y)\right )-\left (i \sin ^{-1}(\lambda y)\right )^n \text {Gamma}\left (1-n,i \sin ^{-1}(\lambda y)\right )\right )}{2 \lambda }\right ) \exp \left (\int _1^y\frac {\left (s \sin ^{-1}(\beta K[1])^k+c \sin ^{-1}\left (\frac {\mu \left (i a \left (\left (-i \sin ^{-1}(\lambda y)\right )^n \text {Gamma}\left (1-n,-i \sin ^{-1}(\lambda y)\right )-\left (i \sin ^{-1}(\lambda y)\right )^n \text {Gamma}\left (1-n,i \sin ^{-1}(\lambda y)\right )\right ) \sin ^{-1}(\lambda y)^{-n}+2 b \lambda x-i a \sin ^{-1}(\lambda K[1])^{-n} \left (\left (-i \sin ^{-1}(\lambda K[1])\right )^n \text {Gamma}\left (1-n,-i \sin ^{-1}(\lambda K[1])\right )-\left (i \sin ^{-1}(\lambda K[1])\right )^n \text {Gamma}\left (1-n,i \sin ^{-1}(\lambda K[1])\right )\right )\right )}{2 b \lambda }\right )^m\right ) \sin ^{-1}(\lambda K[1])^{-n}}{b}dK[1]\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x)+ b*arcsin(lambda*y)^n*diff(w(x,y),y) =(c*arcsin(mu*x)^m+s*arcsin(beta*y)^k)*w(x,y); 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-1 \right ) b\lambda } \left ( a \left ( -\arcsin \left ( \lambda \,y \right ) \LommelS 1 \left ( -n+{\frac {3}{2}},{\frac {1}{2}},\arcsin \left ( \lambda \,y \right ) \right ) + \left ( \arcsin \left ( \lambda \,y \right ) \right ) ^{-n+{\frac {3}{2}}} \right ) \sqrt {-{\lambda }^{2}{y}^{2}+1}-\lambda \, \left ( -ay\LommelS 1 \left ( -n+{\frac {3}{2}},{\frac {1}{2}},\arcsin \left ( \lambda \,y \right ) \right ) +a\LommelS 1 \left ( -n+{\frac {1}{2}},{\frac {3}{2}},\arcsin \left ( \lambda \,y \right ) \right ) ny\arcsin \left ( \lambda \,y \right ) -\sqrt {\arcsin \left ( \lambda \,y \right ) }bx \left ( n-1 \right ) \right ) \right ) {\frac {1}{\sqrt {\arcsin \left ( \lambda \,y \right ) }}}} \right ) {{\rm e}^{\int ^{y}\!{\frac { \left ( \arcsin \left ( {\it \_b}\,\lambda \right ) \right ) ^{-n}}{b} \left ( c \left ( -\arcsin \left ( {\frac { \left ( {\it \_b}\,\lambda +1 \right ) \left ( {\it \_b}\,\lambda -1 \right ) \mu }{ \left ( n-1 \right ) b\lambda \, \left ( {{\it \_b}}^{2}{\lambda }^{2}-1 \right ) } \left ( a{2}^{-n}\arcsin \left ( {\it \_b}\,\lambda \right ) {2}^{n} \left ( -{\LommelS 1 \left ( -n+{\frac {3}{2}},{\frac {1}{2}},\arcsin \left ( {\it \_b}\,\lambda \right ) \right ) {\frac {1}{\sqrt {\arcsin \left ( {\it \_b}\,\lambda \right ) }}}}+ \left ( \arcsin \left ( {\it \_b}\,\lambda \right ) \right ) ^{-n} \right ) \sqrt {-{{\it \_b}}^{2}{\lambda }^{2}+1}+\lambda \, \left ( {a{2}^{-n}{\it \_b}\,{2}^{n}\LommelS 1 \left ( -n+{\frac {3}{2}},{\frac {1}{2}},\arcsin \left ( {\it \_b}\,\lambda \right ) \right ) {\frac {1}{\sqrt {\arcsin \left ( {\it \_b}\,\lambda \right ) }}}}-a{2}^{-n}\sqrt {\arcsin \left ( {\it \_b}\,\lambda \right ) }n{\it \_b}\,{2}^{n}\LommelS 1 \left ( -n+{\frac {1}{2}},{\frac {3}{2}},\arcsin \left ( {\it \_b}\,\lambda \right ) \right ) + \left ( n-1 \right ) \left ( a\int \! \left ( \arcsin \left ( \lambda \,y \right ) \right ) ^{-n}\,{\rm d}y-xb \right ) \right ) \right ) } \right ) \right ) ^{m}+s \left ( \arcsin \left ( \beta \,{\it \_b} \right ) \right ) ^{k} \right ) }{d{\it \_b}}}}\]
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