6.4.20 7.1

   6.4.20.1 [1141] Problem 1
   6.4.20.2 [1142] Problem 2
   6.4.20.3 [1143] Problem 3
   6.4.20.4 [1144] Problem 4
   6.4.20.5 [1145] Problem 5

6.4.20.1 [1141] Problem 1

problem number 1141

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

\[\left \{\left \{w(x,y)\to c_1\left (y-\frac{b x}{a}\right ) \exp \left (\frac{\frac{k \left (\sqrt{a^2 \left (\beta ^2-y^2\right )} (a y-b x) \tan ^{-1}\left (\frac{a y}{\sqrt{a^2 \left (\beta ^2-y^2\right )}}\right )+a^2 \left (\beta ^2-y^2\right )\right )}{b \beta \sqrt{1-\frac{y^2}{\beta ^2}}}+a k x \sin ^{-1}\left (\frac{y}{\beta }\right )+a c \lambda \sqrt{1-\frac{x^2}{\lambda ^2}}+a c x \sin ^{-1}\left (\frac{x}{\lambda }\right )}{a^2}\right )\right \}\right \}\]

Maple

\[w \left ( x,y \right ) ={\it \_F1} \left ({\frac{ya-bx}{a}} \right ){{\rm e}^{{\frac{1}{ab} \left ( a\arcsin \left ({\frac{y}{\beta }} \right ) ky+\sqrt{{\frac{{\beta }^{2}-{y}^{2}}{{\beta }^{2}}}}a\beta \,k+\sqrt{-{\frac{{x}^{2}}{{\lambda }^{2}}}+1}bc\lambda +\arcsin \left ({\frac{x}{\lambda }} \right ) bcx \right ) }}}\]

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6.4.20.2 [1142] Problem 2

problem number 1142

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

\[\left \{\left \{w(x,y)\to c_1\left (y-\frac{b x}{a}\right ) \exp \left (\frac{c \left (\sqrt{-\beta ^2 y^2-2 \beta \lambda x y-\lambda ^2 x^2+1}+(\beta y+\lambda x) \sin ^{-1}(\beta y+\lambda x)\right )}{a \lambda +b \beta }\right )\right \}\right \}\]

Maple

\[w \left ( x,y \right ) ={{\rm e}^{{\frac{ \left ( \sqrt{-{\beta }^{2}{y}^{2}-2\,\beta \,\lambda \,xy-{\lambda }^{2}{x}^{2}+1}+\arcsin \left ( \beta \,y+\lambda \,x \right ) \left ( \beta \,y+\lambda \,x \right ) \right ) c}{a\lambda +b\beta }}}}{\it \_F1} \left ({\frac{ya-bx}{a}} \right ) \]

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6.4.20.3 [1143] Problem 3

problem number 1143

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

\[\left \{\left \{w(x,y)\to c_1\left (y-\frac{b x}{a}\right ) \exp \left (\frac{a \left (\sqrt{-\beta ^2 y^2-2 \beta \lambda x y-\lambda ^2 x^2+1} (-3 a \beta y+a \lambda x+4 b \beta x)+\sin ^{-1}(\beta y+\lambda x) \left (a \left (-2 \beta ^2 y^2+2 \lambda ^2 x^2-1\right )+4 b \beta x (\beta y+\lambda x)\right )\right )}{4 (a \lambda +b \beta )^2}\right )\right \}\right \}\]

Maple

\[w \left ( x,y \right ) ={\it \_F1} \left ({\frac{ya-bx}{a}} \right ){{\rm e}^{1/2\,{\frac{ \left ( \left ( \left ( 1/2\,\lambda \,x-3/2\,\beta \,y \right ) a+2\,bx\beta \right ) \sqrt{-{\beta }^{2}{y}^{2}-2\,\beta \,\lambda \,xy-{\lambda }^{2}{x}^{2}+1}+\arcsin \left ( \beta \,y+\lambda \,x \right ) \left ( \left ({\lambda }^{2}{x}^{2}-{\beta }^{2}{y}^{2}-1/2 \right ) a+2\,b\beta \,x \left ( \beta \,y+\lambda \,x \right ) \right ) \right ) a}{ \left ( a\lambda +b\beta \right ) ^{2}}}}}\]

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6.4.20.4 [1144] Problem 4

problem number 1144

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

\[\left \{\left \{w(x,y)\to c_1\left (y-\int _1^x\frac{b \sin ^{-1}(\lambda K[1])^n}{a}dK[1]\right ) \exp \left (\int _1^x\frac{s \sin ^{-1}\left (\beta \left (y-\int _1^x\frac{b \sin ^{-1}(\lambda K[1])^n}{a}dK[1]+\int _1^{K[2]}\frac{b \sin ^{-1}(\lambda K[1])^n}{a}dK[1]\right )\right ){}^k+c \sin ^{-1}(\mu K[2])^m}{a}dK[2]\right )\right \}\right \}\]

Maple

\[w \left ( x,y \right ) ={\it \_F1} \left ({\frac{-b \left ( -\arcsin \left ( \lambda \,x \right ) \LommelS 1 \left ( n+3/2,1/2,\arcsin \left ( \lambda \,x \right ) \right ) + \left ( \arcsin \left ( \lambda \,x \right ) \right ) ^{n+3/2} \right ) \sqrt{-{\lambda }^{2}{x}^{2}+1}+ \left ( -bx\LommelS 1 \left ( n+3/2,1/2,\arcsin \left ( \lambda \,x \right ) \right ) -\LommelS 1 \left ( n+1/2,3/2,\arcsin \left ( \lambda \,x \right ) \right ) bnx\arcsin \left ( \lambda \,x \right ) +a\sqrt{\arcsin \left ( \lambda \,x \right ) }y \left ( n+1 \right ) \right ) \lambda }{\sqrt{\arcsin \left ( \lambda \,x \right ) }\lambda \,a \left ( n+1 \right ) }} \right ){{\rm e}^{\int ^{x}\!{\frac{1}{a} \left ( c \left ( \arcsin \left ( \mu \,{\it \_b} \right ) \right ) ^{m}+s \left ( -\arcsin \left ({\frac{\beta \, \left ({\it \_b}\,\lambda -1 \right ) \left ({\it \_b}\,\lambda +1 \right ) }{a\lambda \, \left ( n+1 \right ) \left ({{\it \_b}}^{2}{\lambda }^{2}-1 \right ) } \left ( -{2}^{-n}\arcsin \left ({\it \_b}\,\lambda \right ) b{2}^{n} \left ( -{\frac{\LommelS 1 \left ( n+3/2,1/2,\arcsin \left ({\it \_b}\,\lambda \right ) \right ) }{\sqrt{\arcsin \left ({\it \_b}\,\lambda \right ) }}}+ \left ( \arcsin \left ({\it \_b}\,\lambda \right ) \right ) ^{n} \right ) \sqrt{-{{\it \_b}}^{2}{\lambda }^{2}+1}+ \left ( a \left ( n+1 \right ) \int \!{\frac{b \left ( \arcsin \left ( \lambda \,x \right ) \right ) ^{n}}{a}}\,{\rm d}x-{\frac{{2}^{-n}b{\it \_b}\,{2}^{n}\LommelS 1 \left ( n+3/2,1/2,\arcsin \left ({\it \_b}\,\lambda \right ) \right ) }{\sqrt{\arcsin \left ({\it \_b}\,\lambda \right ) }}}-{2}^{-n}b\sqrt{\arcsin \left ({\it \_b}\,\lambda \right ) }\LommelS 1 \left ( n+1/2,3/2,\arcsin \left ({\it \_b}\,\lambda \right ) \right ) n{\it \_b}\,{2}^{n}-ay \left ( n+1 \right ) \right ) \lambda \right ) } \right ) \right ) ^{k} \right ) }{d{\it \_b}}}}\]

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6.4.20.5 [1145] Problem 5

problem number 1145

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

\[\left \{\left \{w(x,y)\to c_1\left (-\frac{b x}{a}-\frac{i \sin ^{-1}(\lambda y)^{-n} \left (\left (-i \sin ^{-1}(\lambda y)\right )^n \text{Gamma}\left (1-n,-i \sin ^{-1}(\lambda y)\right )-\left (i \sin ^{-1}(\lambda y)\right )^n \text{Gamma}\left (1-n,i \sin ^{-1}(\lambda y)\right )\right )}{2 \lambda }\right ) \exp \left (\int _1^y\frac{\left (s \sin ^{-1}(\beta K[1])^k+c \sin ^{-1}\left (\frac{\mu \left (i a \left (\left (-i \sin ^{-1}(\lambda y)\right )^n \text{Gamma}\left (1-n,-i \sin ^{-1}(\lambda y)\right )-\left (i \sin ^{-1}(\lambda y)\right )^n \text{Gamma}\left (1-n,i \sin ^{-1}(\lambda y)\right )\right ) \sin ^{-1}(\lambda y)^{-n}+2 b \lambda x-i a \sin ^{-1}(\lambda K[1])^{-n} \left (\left (-i \sin ^{-1}(\lambda K[1])\right )^n \text{Gamma}\left (1-n,-i \sin ^{-1}(\lambda K[1])\right )-\left (i \sin ^{-1}(\lambda K[1])\right )^n \text{Gamma}\left (1-n,i \sin ^{-1}(\lambda K[1])\right )\right )\right )}{2 b \lambda }\right )^m\right ) \sin ^{-1}(\lambda K[1])^{-n}}{b}dK[1]\right )\right \}\right \}\]

Maple

\[w \left ( x,y \right ) ={\it \_F1} \left ({\frac{a \left ( -\arcsin \left ( y\lambda \right ) \LommelS 1 \left ( -n+3/2,1/2,\arcsin \left ( y\lambda \right ) \right ) + \left ( \arcsin \left ( y\lambda \right ) \right ) ^{-n+3/2} \right ) \sqrt{-{y}^{2}{\lambda }^{2}+1}- \left ( -ay\LommelS 1 \left ( -n+3/2,1/2,\arcsin \left ( y\lambda \right ) \right ) +a\LommelS 1 \left ( -n+1/2,3/2,\arcsin \left ( y\lambda \right ) \right ) ny\arcsin \left ( y\lambda \right ) -\sqrt{\arcsin \left ( y\lambda \right ) }bx \left ( n-1 \right ) \right ) \lambda }{\sqrt{\arcsin \left ( y\lambda \right ) }b\lambda \, \left ( n-1 \right ) }} \right ){{\rm e}^{\int ^{y}\!{\frac{ \left ( \arcsin \left ({\it \_b}\,\lambda \right ) \right ) ^{-n}}{b} \left ( c \left ( -\arcsin \left ({\frac{\mu \, \left ({\it \_b}\,\lambda -1 \right ) \left ({\it \_b}\,\lambda +1 \right ) }{\lambda \,b \left ( n-1 \right ) \left ({{\it \_b}}^{2}{\lambda }^{2}-1 \right ) } \left ( a{2}^{-n}\arcsin \left ({\it \_b}\,\lambda \right ){2}^{n} \left ( -{\frac{\LommelS 1 \left ( -n+3/2,1/2,\arcsin \left ({\it \_b}\,\lambda \right ) \right ) }{\sqrt{\arcsin \left ({\it \_b}\,\lambda \right ) }}}+ \left ( \arcsin \left ({\it \_b}\,\lambda \right ) \right ) ^{-n} \right ) \sqrt{-{{\it \_b}}^{2}{\lambda }^{2}+1}+\lambda \, \left ({\frac{{2}^{-n}a\LommelS 1 \left ( -n+3/2,1/2,\arcsin \left ({\it \_b}\,\lambda \right ) \right ){\it \_b}\,{2}^{n}}{\sqrt{\arcsin \left ({\it \_b}\,\lambda \right ) }}}-a{2}^{-n}\sqrt{\arcsin \left ({\it \_b}\,\lambda \right ) }n{\it \_b}\,{2}^{n}\LommelS 1 \left ( -n+1/2,3/2,\arcsin \left ({\it \_b}\,\lambda \right ) \right ) + \left ( n-1 \right ) \left ( a\int \! \left ( \arcsin \left ( y\lambda \right ) \right ) ^{-n}\,{\rm d}y-bx \right ) \right ) \right ) } \right ) \right ) ^{m}+s \left ( \arcsin \left ( \beta \,{\it \_b} \right ) \right ) ^{k} \right ) }{d{\it \_b}}}}\]

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