88 HFOPDE, chapter 3.7.2

 88.1 Problem 1
 88.2 Problem 2
 88.3 Problem 3
 88.4 Problem 4
 88.5 Problem 5

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88.1 Problem 1

problem number 760

Added Feb. 11, 2019.

Problem Chapter 3.7.2.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b w_y = c \arccos \frac{x}{\lambda }+ k \arccos \frac{y}{\beta } \]

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\frac{cx}{a}\arccos \left ({\frac{x}{\lambda }} \right ) }+{\frac{ky}{b}\arccos \left ({\frac{bx}{a\beta }}+{\frac{ay-bx}{a\beta }} \right ) }+{\frac{1}{ab} \left ( -\sqrt{- \left ({\frac{bx}{a\beta }}+{\frac{ay-bx}{a\beta }} \right ) ^{2}+1}a\beta \,k-\sqrt{-{\frac{{x}^{2}}{{\lambda }^{2}}}+1}bc\lambda +{\it \_F1} \left ({\frac{ay-bx}{a}} \right ) ba \right ) } \]

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88.2 Problem 2

problem number 761

Added Feb. 11, 2019.

Problem Chapter 3.7.2.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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88.3 Problem 3

problem number 762

Added Feb. 11, 2019.

Problem Chapter 3.7.2.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\frac{1}{\beta \,y+\lambda \,x} \left ( -\sqrt{-{\beta }^{2}{y}^{2}-2\,\beta \,\lambda \,xy-{\lambda }^{2}{x}^{2}+1}ax+ \left ( \beta \,y+\lambda \,x \right ) \left ( ax\arccos \left ( \beta \,y+\lambda \,x \right ) +{\it \_F1} \left ({\frac{y}{x}} \right ) \right ) \right ) } \]

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88.4 Problem 4

problem number 763

Added Feb. 11, 2019.

Problem Chapter 3.7.2.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

\[ w \left ( x,y \right ) =\int ^{x}\! \left ( \arccos \left ( \mu \,{\it \_a} \right ) \right ) ^{m}+{\frac{1}{a} \left ( \arccos \left ({\frac{\beta \,\arccos \left ( \lambda \,{\it \_a} \right ) b{\it \_a}}{a}}-{\frac{ \left ( \arccos \left ( \lambda \,x \right ) bx\lambda -\lambda \,ya-\sqrt{-{\lambda }^{2}{x}^{2}+1}b \right ) \beta }{\lambda \,a}}-{\frac{\beta \,\sqrt{-{{\it \_a}}^{2}{\lambda }^{2}+1}b}{\lambda \,a}} \right ) \right ) ^{k}}{d{\it \_a}}+{\it \_F1} \left ({\frac{-\arccos \left ( \lambda \,x \right ) bx\lambda +\lambda \,ya+\sqrt{-{\lambda }^{2}{x}^{2}+1}b}{\lambda \,a}} \right ) \]

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88.5 Problem 5

problem number 764

Added Feb. 11, 2019.

Problem Chapter 3.7.2.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

\[ w \left ( x,y \right ) =\int ^{y}\!{\frac{a}{b\arccos \left ( \lambda \,{\it \_a} \right ) } \left ( \pi -\arccos \left ( -{\frac{\mu \, \left ( x\lambda \,b+a\Si \left ( \arccos \left ( \lambda \,y \right ) \right ) \right ) }{\lambda \,b}}+{\frac{\mu \,a\Si \left ( \arccos \left ( \lambda \,{\it \_a} \right ) \right ) }{\lambda \,b}} \right ) \right ) ^{m}}+{\frac{ \left ( \arccos \left ({\it \_a}\,\beta \right ) \right ) ^{k}}{b\arccos \left ( \lambda \,{\it \_a} \right ) }}{d{\it \_a}}+{\it \_F1} \left ({\frac{x\lambda \,b+a\Si \left ( \arccos \left ( \lambda \,y \right ) \right ) }{\lambda \,b}} \right ) \]