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'));
\[w \left (x , y\right ) = \mathit {\_F1} \left (\frac {a y -b x}{a}\right ) {\mathrm e}^{\frac {a k y \arcsin \left (\frac {y}{\beta }\right )+b c x \arcsin \left (\frac {x}{\lambda }\right )+\sqrt {\frac {\beta ^{2}-y^{2}}{\beta ^{2}}}\, a \beta k +\sqrt {-\frac {x^{2}}{\lambda ^{2}}+1}\, b c \lambda }{a b}}\]
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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'));
\[w \left (x , y\right ) = \mathit {\_F1} \left (\frac {a y -b x}{a}\right ) {\mathrm e}^{\frac {\left (\left (\beta y +\lambda x \right ) \arcsin \left (\beta y +\lambda x \right )+\sqrt {-\beta ^{2} y^{2}-2 \beta \lambda x y -\lambda ^{2} x^{2}+1}\right ) c}{a \lambda +b \beta }}\]
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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'));
\[w \left (x , y\right ) = \mathit {\_F1} \left (\frac {a y -b x}{a}\right ) {\mathrm e}^{\frac {\left (\left (2 \left (\beta y +\lambda x \right ) b \beta x +\left (-\beta ^{2} y^{2}+\lambda ^{2} x^{2}-\frac {1}{2}\right ) a \right ) \arcsin \left (\beta y +\lambda x \right )+\left (2 b \beta x +\left (-\frac {3 \beta y}{2}+\frac {\lambda x}{2}\right ) a \right ) \sqrt {-\beta ^{2} y^{2}-2 \beta \lambda x y -\lambda ^{2} x^{2}+1}\right ) a}{2 \left (a \lambda +b \beta \right )^{2}}}\]
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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'));
\[w \left (x , y\right ) = \mathit {\_F1} \left (\frac {-\left (-\LommelS 1\left (n +\frac {3}{2}, \frac {1}{2}, \arcsin \left (\lambda x \right )\right ) \arcsin \left (\lambda x \right )+\arcsin \left (\lambda x \right )^{n +\frac {3}{2}}\right ) \sqrt {-\lambda ^{2} x^{2}+1}\, b +\left (-b n x \LommelS 1\left (n +\frac {1}{2}, \frac {3}{2}, \arcsin \left (\lambda x \right )\right ) \arcsin \left (\lambda x \right )-b x \LommelS 1\left (n +\frac {3}{2}, \frac {1}{2}, \arcsin \left (\lambda x \right )\right )+\left (n +1\right ) a y \sqrt {\arcsin \left (\lambda x \right )}\right ) \lambda }{\left (n +1\right ) a \lambda \sqrt {\arcsin \left (\lambda x \right )}}\right ) {\mathrm e}^{\int _{}^{x}\frac {c \arcsin \left (\mathit {\_b} \mu \right )^{m}+s \left (-\arcsin \left (\frac {\left (\mathit {\_b} \lambda -1\right ) \left (\mathit {\_b} \lambda +1\right ) \left (-\left (\arcsin \left (\mathit {\_b} \lambda \right )^{n}-\frac {\LommelS 1\left (n +\frac {3}{2}, \frac {1}{2}, \arcsin \left (\mathit {\_b} \lambda \right )\right )}{\sqrt {\arcsin \left (\mathit {\_b} \lambda \right )}}\right ) \sqrt {-\mathit {\_b}^{2} \lambda ^{2}+1}\, b 2^{n} 2^{-n} \arcsin \left (\mathit {\_b} \lambda \right )+\left (-\mathit {\_b} b n 2^{n} 2^{-n} \LommelS 1\left (n +\frac {1}{2}, \frac {3}{2}, \arcsin \left (\mathit {\_b} \lambda \right )\right ) \sqrt {\arcsin \left (\mathit {\_b} \lambda \right )}-\frac {\mathit {\_b} b 2^{n} 2^{-n} \LommelS 1\left (n +\frac {3}{2}, \frac {1}{2}, \arcsin \left (\mathit {\_b} \lambda \right )\right )}{\sqrt {\arcsin \left (\mathit {\_b} \lambda \right )}}-\left (n +1\right ) a y +\left (n +1\right ) a \left (\int \frac {b \arcsin \left (\lambda x \right )^{n}}{a}d x \right )\right ) \lambda \right ) \beta }{\left (n +1\right ) \left (\mathit {\_b}^{2} \lambda ^{2}-1\right ) a \lambda }\right )\right )^{k}}{a}d\mathit {\_b}}\]
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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]];
\[\left \{\left \{w(x,y)\to c_1\left (-\frac {b x}{a}-\frac {i \sin ^{-1}(\lambda y)^{-n} \left (\left (-i \sin ^{-1}(\lambda y)\right )^n \operatorname {Gamma}\left (1-n,-i \sin ^{-1}(\lambda y)\right )-\left (i \sin ^{-1}(\lambda y)\right )^n \operatorname {Gamma}\left (1-n,i \sin ^{-1}(\lambda y)\right )\right )}{2 \lambda }\right ) \exp \left (\int _1^y\frac {\left (s \sin ^{-1}(\beta K[1])^k+c \sin ^{-1}\left (\frac {\mu \left (i a \left (\left (-i \sin ^{-1}(\lambda y)\right )^n \operatorname {Gamma}\left (1-n,-i \sin ^{-1}(\lambda y)\right )-\left (i \sin ^{-1}(\lambda y)\right )^n \operatorname {Gamma}\left (1-n,i \sin ^{-1}(\lambda y)\right )\right ) \sin ^{-1}(\lambda y)^{-n}+2 b \lambda x-i a \sin ^{-1}(\lambda K[1])^{-n} \left (\left (-i \sin ^{-1}(\lambda K[1])\right )^n \operatorname {Gamma}\left (1-n,-i \sin ^{-1}(\lambda K[1])\right )-\left (i \sin ^{-1}(\lambda K[1])\right )^n \operatorname {Gamma}\left (1-n,i \sin ^{-1}(\lambda K[1])\right )\right )\right )}{2 b \lambda }\right )^m\right ) \sin ^{-1}(\lambda K[1])^{-n}}{b}dK[1]\right )\right \}\right \}\]
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'));
\[w \left (x , y\right ) = \mathit {\_F1} \left (\frac {\left (-\LommelS 1\left (-n +\frac {3}{2}, \frac {1}{2}, \arcsin \left (\lambda y \right )\right ) \arcsin \left (\lambda y \right )+\arcsin \left (\lambda y \right )^{-n +\frac {3}{2}}\right ) \sqrt {-\lambda ^{2} y^{2}+1}\, a -\left (a n y \LommelS 1\left (-n +\frac {1}{2}, \frac {3}{2}, \arcsin \left (\lambda y \right )\right ) \arcsin \left (\lambda y \right )-a y \LommelS 1\left (-n +\frac {3}{2}, \frac {1}{2}, \arcsin \left (\lambda y \right )\right )-\left (n -1\right ) b x \sqrt {\arcsin \left (\lambda y \right )}\right ) \lambda }{\left (n -1\right ) b \lambda \sqrt {\arcsin \left (\lambda y \right )}}\right ) {\mathrm e}^{\int _{}^{y}\frac {\left (c \left (-\arcsin \left (\frac {\left (\mathit {\_b} \lambda -1\right ) \left (\mathit {\_b} \lambda +1\right ) \left (\left (\arcsin \left (\mathit {\_b} \lambda \right )^{-n}-\frac {\LommelS 1\left (-n +\frac {3}{2}, \frac {1}{2}, \arcsin \left (\mathit {\_b} \lambda \right )\right )}{\sqrt {\arcsin \left (\mathit {\_b} \lambda \right )}}\right ) \sqrt {-\mathit {\_b}^{2} \lambda ^{2}+1}\, a 2^{n} 2^{-n} \arcsin \left (\mathit {\_b} \lambda \right )+\left (-\mathit {\_b} a n 2^{n} 2^{-n} \LommelS 1\left (-n +\frac {1}{2}, \frac {3}{2}, \arcsin \left (\mathit {\_b} \lambda \right )\right ) \sqrt {\arcsin \left (\mathit {\_b} \lambda \right )}+\frac {\mathit {\_b} a 2^{n} 2^{-n} \LommelS 1\left (-n +\frac {3}{2}, \frac {1}{2}, \arcsin \left (\mathit {\_b} \lambda \right )\right )}{\sqrt {\arcsin \left (\mathit {\_b} \lambda \right )}}+\left (n -1\right ) \left (a \left (\int \arcsin \left (\lambda y \right )^{-n}d y \right )-b x \right )\right ) \lambda \right ) \mu }{\left (n -1\right ) \left (\mathit {\_b}^{2} \lambda ^{2}-1\right ) b \lambda }\right )\right )^{m}+s \arcsin \left (\mathit {\_b} \beta \right )^{k}\right ) \arcsin \left (\mathit {\_b} \lambda \right )^{-n}}{b}d\mathit {\_b}}\]
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