115 HFOPDE, chapter 4.7.2

 115.1 Problem 1
 115.2 Problem 2
 115.3 Problem 3
 115.4 Problem 4
 115.5 Problem 5

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115.1 Problem 1

problem number 938

Added March 9, 2019.

Problem Chapter 4.7.2.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ a w_x + b w_y = \left ( c \arccos (\frac{x}{\lambda } + k \arccos (\frac{y}{\beta } ) \right ) w \]

Mathematica

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

Maple

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

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115.2 Problem 2

problem number 939

Added March 9, 2019.

Problem Chapter 4.7.2.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ a w_x + b w_y = c \arccos (\lambda x+\beta y) w \]

Mathematica

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

Maple

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

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115.3 Problem 3

problem number 940

Added March 9, 2019.

Problem Chapter 4.7.2.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ a w_x + b w_y = a x \arccos (\lambda x+\beta y) w \]

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({\frac{ay-xb}{a}} \right ){{\rm e}^{1/4\,{\frac{a \left ( -2\,\arccos \left ( \beta \,y+\lambda \,x \right ) a{\beta }^{2}{y}^{2}+2\,\arccos \left ( \beta \,y+\lambda \,x \right ) a{\lambda }^{2}{x}^{2}+4\,\arccos \left ( \beta \,y+\lambda \,x \right ) b{\beta }^{2}xy+4\,\arccos \left ( \beta \,y+\lambda \,x \right ) b\beta \,\lambda \,{x}^{2}+3\,\sqrt{-{\beta }^{2}{y}^{2}-2\,\beta \,\lambda \,xy-{\lambda }^{2}{x}^{2}+1}a\beta \,y-\sqrt{-{\beta }^{2}{y}^{2}-2\,\beta \,\lambda \,xy-{\lambda }^{2}{x}^{2}+1}a\lambda \,x-4\,\sqrt{-{\beta }^{2}{y}^{2}-2\,\beta \,\lambda \,xy-{\lambda }^{2}{x}^{2}+1}b\beta \,x+\arcsin \left ( \beta \,y+\lambda \,x \right ) a \right ) }{ \left ( a\lambda +b\beta \right ) ^{2}}}}} \]

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115.4 Problem 4

problem number 941

Added March 9, 2019.

Problem Chapter 4.7.2.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ a w_x + b \arccos ^n(\lambda x)w_y = \left ( c \arccos ^m(\mu x) + s \arccos ^k(\beta y) \right ) w \]

Mathematica

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

Maple

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

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115.5 Problem 5

problem number 942

Added March 9, 2019.

Problem Chapter 4.7.2.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ a w_x + b \arccos ^n(\lambda y)w_y = \left ( c \arccos ^m(\mu x) + s \arccos ^k(\beta y) \right ) w \]

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({\frac{a{2}^{-n}\sqrt{\pi }}{\lambda \,b} \left ( -{\frac{ \left ( \arccos \left ( \lambda \,y \right ) \right ) ^{-n+1}{2}^{n}\sqrt{-{\lambda }^{2}{y}^{2}+1}}{\sqrt{\pi } \left ( n-2 \right ) }}+{\frac{{2}^{n}\sqrt{\arccos \left ( \lambda \,y \right ) }\LommelS 1 \left ( -n+3/2,3/2,\arccos \left ( \lambda \,y \right ) \right ) \sqrt{-{\lambda }^{2}{y}^{2}+1}}{\sqrt{\pi } \left ( n-2 \right ) }}+3/2\,{\frac{{2}^{n} \left ( -2/3\,n+4/3 \right ) \left ( \lambda \,y\arccos \left ( \lambda \,y \right ) -\sqrt{-{\lambda }^{2}{y}^{2}+1} \right ) \LommelS 1 \left ( -n+1/2,1/2,\arccos \left ( \lambda \,y \right ) \right ) }{\sqrt{\pi } \left ( n-2 \right ) \sqrt{\arccos \left ( \lambda \,y \right ) }}} \right ) }+x \right ){{\rm e}^{\int ^{y}\!{\frac{ \left ( \arccos \left ( \lambda \,{\it \_b} \right ) \right ) ^{-n}}{b} \left ( c \left ( \arccos \left ( \mu \, \left ( -{\frac{a{2}^{-n}\sqrt{\pi }}{\lambda \,b} \left ( -{\frac{ \left ( \arccos \left ( \lambda \,{\it \_b} \right ) \right ) ^{-n+1}{2}^{n}\sqrt{-{{\it \_b}}^{2}{\lambda }^{2}+1}}{\sqrt{\pi } \left ( n-2 \right ) }}+{\frac{{2}^{n}\sqrt{\arccos \left ( \lambda \,{\it \_b} \right ) }\LommelS 1 \left ( -n+3/2,3/2,\arccos \left ( \lambda \,{\it \_b} \right ) \right ) \sqrt{-{{\it \_b}}^{2}{\lambda }^{2}+1}}{\sqrt{\pi } \left ( n-2 \right ) }}+3/2\,{\frac{{2}^{n} \left ( -2/3\,n+4/3 \right ) \left ( \lambda \,{\it \_b}\,\arccos \left ( \lambda \,{\it \_b} \right ) -\sqrt{-{{\it \_b}}^{2}{\lambda }^{2}+1} \right ) \LommelS 1 \left ( -n+1/2,1/2,\arccos \left ( \lambda \,{\it \_b} \right ) \right ) }{\sqrt{\pi } \left ( n-2 \right ) \sqrt{\arccos \left ( \lambda \,{\it \_b} \right ) }}} \right ) }-{\frac{a\int \! \left ( \arccos \left ( \lambda \,y \right ) \right ) ^{-n}\,{\rm d}y}{b}}+x \right ) \right ) \right ) ^{m}+s \left ( \arccos \left ( \beta \,{\it \_b} \right ) \right ) ^{k} \right ) }{d{\it \_b}}}} \]