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'));
\[w \left ( x,y \right ) ={\frac {1}{ab} \left ( \sqrt {-{\frac {{x}^{2}}{{\lambda }^{2}}}+1}bc\lambda +{\it \_F1} \left ( {\frac {ay-xb}{a}} \right ) ba+\arcsin \left ( {\frac {x}{\lambda }} \right ) bcx+ka \left ( \sqrt {{\frac {{\beta }^{2}-{y}^{2}}{{\beta }^{2}}}}\beta +\arcsin \left ( {\frac {y}{\beta }} \right ) y \right ) \right ) }\]
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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'));
\[w \left ( x,y \right ) ={\frac {1}{a\lambda +\beta \,b} \left ( \sqrt {-{\beta }^{2}{y}^{2}-2\,\beta \,\lambda \,xy-{x}^{2}{\lambda }^{2}+1}c+ \left ( a\lambda +\beta \,b \right ) {\it \_F1} \left ( {\frac {ay-xb}{a}} \right ) +c\arcsin \left ( \beta \,y+x\lambda \right ) \left ( \beta \,y+x\lambda \right ) \right ) }\]
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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'));
\[w \left ( x,y \right ) ={\frac {1}{\beta \,y+x\lambda } \left ( \sqrt {-{x}^{2} \left ( {\frac {\beta \,y}{x}}+\lambda \right ) ^{2}+1}ax+ \left ( \beta \,y+x\lambda \right ) \left ( ax\arcsin \left ( \beta \,y+x\lambda \right ) +{\it \_F1} \left ( {\frac {y}{x}} \right ) \right ) \right ) }\]
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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]];
\[\left \{\left \{w(x,y)\to \int _1^x\left (\frac {\sin ^{-1}\left (\frac {\beta \left (i b \sin ^{-1}(\lambda x)^n \left (\left (i \sin ^{-1}(\lambda x)\right )^n \text {Gamma}\left (n+1,-i \sin ^{-1}(\lambda x)\right )-\left (-i \sin ^{-1}(\lambda x)\right )^n \text {Gamma}\left (n+1,i \sin ^{-1}(\lambda x)\right )\right ) \left (\sin ^{-1}(\lambda x)^2\right )^{-n}+2 a \lambda y-i b \sin ^{-1}(\lambda K[1])^n \left (\sin ^{-1}(\lambda K[1])^2\right )^{-n} \left (\left (i \sin ^{-1}(\lambda K[1])\right )^n \text {Gamma}\left (n+1,-i \sin ^{-1}(\lambda K[1])\right )-\left (-i \sin ^{-1}(\lambda K[1])\right )^n \text {Gamma}\left (n+1,i \sin ^{-1}(\lambda K[1])\right )\right )\right )}{2 a \lambda }\right )^k}{a}+\sin ^{-1}(\mu K[1])^m\right )dK[1]+c_1\left (y+\frac {i b \sin ^{-1}(\lambda x)^n \left (\sin ^{-1}(\lambda x)^2\right )^{-n} \left (\left (i \sin ^{-1}(\lambda x)\right )^n \text {Gamma}\left (n+1,-i \sin ^{-1}(\lambda x)\right )-\left (-i \sin ^{-1}(\lambda x)\right )^n \text {Gamma}\left (n+1,i \sin ^{-1}(\lambda x)\right )\right )}{2 a \lambda }\right )\right \}\right \}\]
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'));
\[w \left ( x,y \right ) =\int ^{x}\! \left ( \arcsin \left ( \mu \,{\it \_a} \right ) \right ) ^{m}+{\frac {1}{a} \left ( \arcsin \left ( {\frac {\beta }{a\lambda } \left ( \sqrt {-{{\it \_a}}^{2}{\lambda }^{2}+1}b-\sqrt {-{x}^{2}{\lambda }^{2}+1}b+\lambda \, \left ( -\arcsin \left ( x\lambda \right ) bx+b{\it \_a}\,\arcsin \left ( {\it \_a}\,\lambda \right ) +ay \right ) \right ) } \right ) \right ) ^{k}}{d{\it \_a}}+{\it \_F1} \left ( {\frac {1}{a\lambda } \left ( -\arcsin \left ( x\lambda \right ) bx\lambda +ya\lambda -\sqrt {-{x}^{2}{\lambda }^{2}+1}b \right ) } \right ) \]
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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]];
\[\left \{\left \{w(x,y)\to \int _1^y\frac {\left (\sin ^{-1}(\beta K[1])^k+a \sin ^{-1}\left (\frac {\mu \left (i a \left (\left (-i \sin ^{-1}(\lambda y)\right )^n \text {Gamma}\left (1-n,-i \sin ^{-1}(\lambda y)\right )-\left (i \sin ^{-1}(\lambda y)\right )^n \text {Gamma}\left (1-n,i \sin ^{-1}(\lambda y)\right )\right ) \sin ^{-1}(\lambda y)^{-n}+2 b \lambda x-i a \sin ^{-1}(\lambda K[1])^{-n} \left (\left (-i \sin ^{-1}(\lambda K[1])\right )^n \text {Gamma}\left (1-n,-i \sin ^{-1}(\lambda K[1])\right )-\left (i \sin ^{-1}(\lambda K[1])\right )^n \text {Gamma}\left (1-n,i \sin ^{-1}(\lambda K[1])\right )\right )\right )}{2 b \lambda }\right )^m\right ) \sin ^{-1}(\lambda K[1])^{-n}}{b}dK[1]+c_1\left (-\frac {b x}{a}-\frac {i \sin ^{-1}(\lambda y)^{-n} \left (\left (-i \sin ^{-1}(\lambda y)\right )^n \text {Gamma}\left (1-n,-i \sin ^{-1}(\lambda y)\right )-\left (i \sin ^{-1}(\lambda y)\right )^n \text {Gamma}\left (1-n,i \sin ^{-1}(\lambda y)\right )\right )}{2 \lambda }\right )\right \}\right \}\]
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'));
\[w \left ( x,y \right ) =\int ^{y}\!{\frac {1}{\arcsin \left ( {\it \_a}\,\lambda \right ) b} \left ( a \left ( -\arcsin \left ( {\frac {\mu \, \left ( -xb\lambda +\Ci \left ( \arcsin \left ( \lambda \,y \right ) \right ) a-\Ci \left ( \arcsin \left ( {\it \_a}\,\lambda \right ) \right ) a \right ) }{b\lambda }} \right ) \right ) ^{m}+ \left ( \arcsin \left ( \beta \,{\it \_a} \right ) \right ) ^{k} \right ) }{d{\it \_a}}+{\it \_F1} \left ( {\frac {xb\lambda -\Ci \left ( \arcsin \left ( \lambda \,y \right ) \right ) a}{b\lambda }} \right ) \]
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