117 HFOPDE, chapter 4.7.4

 117.1 Problem 1
 117.2 Problem 2
 117.3 Problem 3
 117.4 Problem 4
 117.5 Problem 5

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117.1 Problem 1

problem number 948

Added March 9, 2019.

Problem Chapter 4.7.4.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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117.2 Problem 2

problem number 949

Added March 9, 2019.

Problem Chapter 4.7.4.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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117.3 Problem 3

problem number 950

Added March 9, 2019.

Problem Chapter 4.7.4.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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117.4 Problem 4

problem number 951

Added March 9, 2019.

Problem Chapter 4.7.4.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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117.5 Problem 5

problem number 952

Added March 9, 2019.

Problem Chapter 4.7.4.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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