116 HFOPDE, chapter 4.7.3

 116.1 Problem 1
 116.2 Problem 2
 116.3 Problem 3
 116.4 Problem 4
 116.5 Problem 5

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116.1 Problem 1

problem number 943

Added March 9, 2019.

Problem Chapter 4.7.3.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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116.2 Problem 2

problem number 944

Added March 9, 2019.

Problem Chapter 4.7.3.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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116.3 Problem 3

problem number 945

Added March 9, 2019.

Problem Chapter 4.7.3.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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116.4 Problem 4

problem number 946

Added March 9, 2019.

Problem Chapter 4.7.3.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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116.5 Problem 5

problem number 947

Added March 9, 2019.

Problem Chapter 4.7.3.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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