114 HFOPDE, chapter 4.7.1

 114.1 Problem 1
 114.2 Problem 2
 114.3 Problem 3
 114.4 Problem 4
 114.5 Problem 5

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114.1 Problem 1

problem number 933

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

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

Maple

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

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114.2 Problem 2

problem number 934

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

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

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114.3 Problem 3

problem number 935

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

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

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114.4 Problem 4

problem number 936

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

\[ \text{Timed out} \] Timed out

Maple

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

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114.5 Problem 5

problem number 937

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

\[ \text{Timed out} \] Timed out

Maple

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