Added Feb. 11, 2019.
Problem Chapter 3.7.1.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b w_y = c \arcsin \frac {x}{\lambda }+ k \arcsin \frac {y}{\beta } \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*D[w[x, y], y] == c*ArcSin[x/lambda] + k*ArcSin[y/beta]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \frac {a^2 b \beta c_1\left (y-\frac {b x}{a}\right )-\frac {b k x \sqrt {a^2 \left (\beta ^2-y^2\right )} \tan ^{-1}\left (\frac {a y}{\sqrt {a^2 \left (\beta ^2-y^2\right )}}\right )}{\sqrt {1-\frac {y^2}{\beta ^2}}}-\frac {a^2 k y^2}{\sqrt {1-\frac {y^2}{\beta ^2}}}+\frac {a^2 \beta ^2 k}{\sqrt {1-\frac {y^2}{\beta ^2}}}+\frac {a k y \sqrt {a^2 \left (\beta ^2-y^2\right )} \tan ^{-1}\left (\frac {a y}{\sqrt {a^2 \left (\beta ^2-y^2\right )}}\right )}{\sqrt {1-\frac {y^2}{\beta ^2}}}+a b \beta c \lambda \sqrt {1-\frac {x^2}{\lambda ^2}}+a b \beta c x \sin ^{-1}\left (\frac {x}{\lambda }\right )+a b \beta k x \sin ^{-1}\left (\frac {y}{\beta }\right )}{a^2 b \beta }\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x) + b*diff(w(x,y),y) = c*arcsin(x/lambda)+k*arcsin(y/beta); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[w \left (x , y\right ) = \frac {b c x \arcsin \left (\frac {x}{\lambda }\right )+a b \textit {\_F1} \left (\frac {a y -b x}{a}\right )+\sqrt {-\frac {x^{2}}{\lambda ^{2}}+1}\, b c \lambda +\left (y \arcsin \left (\frac {y}{\beta }\right )+\sqrt {\frac {\beta ^{2}-y^{2}}{\beta ^{2}}}\, \beta \right ) a k}{a b}\]
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Added Feb. 11, 2019.
Problem Chapter 3.7.1.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b w_y = c \arcsin (\lambda x+\beta y) \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*D[w[x, y], y] == c*ArcSin[lambda*x + beta*y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \frac {c \left (\sqrt {-\beta ^2 y^2-2 \beta \lambda x y-\lambda ^2 x^2+1}+(\beta y+\lambda x) \sin ^{-1}(\beta y+\lambda x)\right )}{a \lambda +b \beta }+c_1\left (y-\frac {b x}{a}\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x) + b*diff(w(x,y),y) = c *arcsin(lambda*x+beta*y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[w \left (x , y\right ) = \frac {\left (\beta y +\lambda x \right ) c \arcsin \left (\beta y +\lambda x \right )+\sqrt {-\beta ^{2} y^{2}-2 \beta \lambda x y -\lambda ^{2} x^{2}+1}\, c +\left (a \lambda +b \beta \right ) \textit {\_F1} \left (\frac {a y -b x}{a}\right )}{a \lambda +b \beta }\]
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Added Feb. 11, 2019.
Problem Chapter 3.7.1.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ x w_x + y w_y = a x \arcsin (\lambda x+\beta y) \]
Mathematica ✓
ClearAll["Global`*"]; pde = x*D[w[x, y], x] + y*D[w[x, y], y] == a*x*ArcSin[lambda*x + beta*y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to a x \left (\frac {\sqrt {-\beta ^2 y^2-2 \beta \lambda x y-\lambda ^2 x^2+1}}{\beta y+\lambda x}+\sin ^{-1}(\beta y+\lambda x)\right )+c_1\left (\frac {y}{x}\right )\right \}\right \}\]
Maple ✓
restart; pde := x*diff(w(x,y),x) + y*diff(w(x,y),y) = a*x *arcsin(lambda*x+beta*y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[w \left (x , y\right ) = \frac {\sqrt {-\left (\frac {\beta y}{x}+\lambda \right )^{2} x^{2}+1}\, a x +\left (\beta y +\lambda x \right ) \left (a x \arcsin \left (\beta y +\lambda x \right )+\textit {\_F1} \left (\frac {y}{x}\right )\right )}{\beta y +\lambda x}\]
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Added Feb. 11, 2019.
Problem Chapter 3.7.1.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b \arcsin ^n(\lambda x) w_y = c \arcsin ^m(\mu x)+s \arcsin ^k(\beta y) \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*ArcSin[lambda*x]^n*D[w[x, y], y] == a*ArcSin[mu*x]^m + ArcSin[beta*y]^k; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \int _1^x\left (\frac {\sin ^{-1}\left (\frac {\beta \left (i b \sin ^{-1}(\lambda x)^n \left (\left (i \sin ^{-1}(\lambda x)\right )^n \operatorname {Gamma}\left (n+1,-i \sin ^{-1}(\lambda x)\right )-\left (-i \sin ^{-1}(\lambda x)\right )^n \operatorname {Gamma}\left (n+1,i \sin ^{-1}(\lambda x)\right )\right ) \left (\sin ^{-1}(\lambda x)^2\right )^{-n}+2 a \lambda y-i b \sin ^{-1}(\lambda K[1])^n \left (\sin ^{-1}(\lambda K[1])^2\right )^{-n} \left (\left (i \sin ^{-1}(\lambda K[1])\right )^n \operatorname {Gamma}\left (n+1,-i \sin ^{-1}(\lambda K[1])\right )-\left (-i \sin ^{-1}(\lambda K[1])\right )^n \operatorname {Gamma}\left (n+1,i \sin ^{-1}(\lambda K[1])\right )\right )\right )}{2 a \lambda }\right )^k}{a}+\sin ^{-1}(\mu K[1])^m\right )dK[1]+c_1\left (y+\frac {i b \sin ^{-1}(\lambda x)^n \left (\sin ^{-1}(\lambda x)^2\right )^{-n} \left (\left (i \sin ^{-1}(\lambda x)\right )^n \operatorname {Gamma}\left (n+1,-i \sin ^{-1}(\lambda x)\right )-\left (-i \sin ^{-1}(\lambda x)\right )^n \operatorname {Gamma}\left (n+1,i \sin ^{-1}(\lambda x)\right )\right )}{2 a \lambda }\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x) + b*arcsin(lambda*x)*diff(w(x,y),y) = a*arcsin(mu*x)^m+arcsin(beta*y)^k; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[w \left (x , y\right ) = \int _{}^{x}\left (\arcsin \left (\textit {\_a} \mu \right )^{m}+\frac {\arcsin \left (\frac {\left (\sqrt {-\textit {\_a}^{2} \lambda ^{2}+1}\, b -\sqrt {-\lambda ^{2} x^{2}+1}\, b +\left (\textit {\_a} b \arcsin \left (\textit {\_a} \lambda \right )-b x \arcsin \left (\lambda x \right )+a y \right ) \lambda \right ) \beta }{a \lambda }\right )^{k}}{a}\right )d \textit {\_a} +\textit {\_F1} \left (\frac {-b \lambda x \arcsin \left (\lambda x \right )+a \lambda y -\sqrt {-\lambda ^{2} x^{2}+1}\, b}{a \lambda }\right )\]
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Added Feb. 11, 2019.
Problem Chapter 3.7.1.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b \arcsin ^n(\lambda y) w_y = c \arcsin ^m(\mu x)+s \arcsin ^k(\beta y) \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*ArcSin[lambda*y]^n*D[w[x, y], y] == a*ArcSin[mu*x]^m + ArcSin[beta*y]^k; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \int _1^y\frac {\left (\sin ^{-1}(\beta K[1])^k+a \sin ^{-1}\left (\frac {\mu \left (i a \left (\left (-i \sin ^{-1}(\lambda y)\right )^n \operatorname {Gamma}\left (1-n,-i \sin ^{-1}(\lambda y)\right )-\left (i \sin ^{-1}(\lambda y)\right )^n \operatorname {Gamma}\left (1-n,i \sin ^{-1}(\lambda y)\right )\right ) \sin ^{-1}(\lambda y)^{-n}+2 b \lambda x-i a \sin ^{-1}(\lambda K[1])^{-n} \left (\left (-i \sin ^{-1}(\lambda K[1])\right )^n \operatorname {Gamma}\left (1-n,-i \sin ^{-1}(\lambda K[1])\right )-\left (i \sin ^{-1}(\lambda K[1])\right )^n \operatorname {Gamma}\left (1-n,i \sin ^{-1}(\lambda K[1])\right )\right )\right )}{2 b \lambda }\right )^m\right ) \sin ^{-1}(\lambda K[1])^{-n}}{b}dK[1]+c_1\left (-\frac {b x}{a}-\frac {i \sin ^{-1}(\lambda y)^{-n} \left (\left (-i \sin ^{-1}(\lambda y)\right )^n \operatorname {Gamma}\left (1-n,-i \sin ^{-1}(\lambda y)\right )-\left (i \sin ^{-1}(\lambda y)\right )^n \operatorname {Gamma}\left (1-n,i \sin ^{-1}(\lambda y)\right )\right )}{2 \lambda }\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x) + b*arcsin(lambda*y)*diff(w(x,y),y) = a*arcsin(mu*x)^m+arcsin(beta*y)^k; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
\[w \left (x , y\right ) = \int _{}^{y}\frac {a \left (-\arcsin \left (\frac {\left (-b \lambda x -a \Ci \left (\arcsin \left (\textit {\_a} \lambda \right )\right )+a \Ci \left (\arcsin \left (\lambda y \right )\right )\right ) \mu }{b \lambda }\right )\right )^{m}+\arcsin \left (\textit {\_a} \beta \right )^{k}}{b \arcsin \left (\textit {\_a} \lambda \right )}d \textit {\_a} +\textit {\_F1} \left (\frac {b \lambda x -a \Ci \left (\arcsin \left (\lambda y \right )\right )}{b \lambda }\right )\]
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