89 HFOPDE, chapter 3.7.3

 89.1 Problem 1
 89.2 Problem 2
 89.3 Problem 3
 89.4 Problem 4
 89.5 Problem 5

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89.1 Problem 1

problem number 765

Added Feb. 11, 2019.

Problem Chapter 3.7.3.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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89.2 Problem 2

problem number 766

Added Feb. 11, 2019.

Problem Chapter 3.7.3.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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89.3 Problem 3

problem number 767

Added Feb. 11, 2019.

Problem Chapter 3.7.3.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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89.4 Problem 4

problem number 768

Added Feb. 11, 2019.

Problem Chapter 3.7.3.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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89.5 Problem 5

problem number 769

Added Feb. 11, 2019.

Problem Chapter 3.7.3.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

\[ w \left ( x,y \right ) =\int ^{y}\!{\frac{a}{b\arctan \left ( \lambda \,{\it \_b} \right ) } \left ( \arctan \left ({\frac{\mu \,a\int \! \left ( \arctan \left ( \lambda \,{\it \_b} \right ) \right ) ^{-1}\,{\rm d}{\it \_b}}{b}}+\mu \, \left ( -\int \!{\frac{a}{b\arctan \left ( \lambda \,y \right ) }}\,{\rm d}y+x \right ) \right ) \right ) ^{m}}+{\frac{ \left ( \arctan \left ( \beta \,{\it \_b} \right ) \right ) ^{k}}{b\arctan \left ( \lambda \,{\it \_b} \right ) }}{d{\it \_b}}+{\it \_F1} \left ( -\int \!{\frac{a}{b\arctan \left ( \lambda \,y \right ) }}\,{\rm d}y+x \right ) \]