112 HFOPDE, chapter 4.6.4

 112.1 Problem 1
 112.2 Problem 2
 112.3 Problem 3
 112.4 Problem 4
 112.5 Problem 5

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112.1 Problem 1

problem number 922

Added March 9, 2019.

Problem Chapter 4.6.4.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({\frac{ay-bx}{a}} \right ) \left ( \left ( \cot \left ({\frac{ \left ( ay-bx \right ) \mu +\lambda \,xa+b\mu \,x}{a}} \right ) \right ) ^{2}+1 \right ) ^{-1/2\,{\frac{c}{a\lambda +b\mu }}} \]

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112.2 Problem 2

problem number 923

Added March 9, 2019.

Problem Chapter 4.6.4.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b w_y = (c \cot (\lambda x)+ k \cot (\mu y) ) w \]

Mathematica

\[ \left \{\left \{w(x,y)\to c_1\left (y-\frac{b x}{a}\right ) \sin ^{\frac{c}{a \lambda }}(\lambda x) \sin ^{\frac{k}{b \mu }}(\mu y)\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({\frac{ay-bx}{a}} \right ) \left ( \left ( \cot \left ( \lambda \,x \right ) \right ) ^{2}+1 \right ) ^{-1/2\,{\frac{c}{a\lambda }}} \left ( \left ( \cot \left ( \mu \,y \right ) \right ) ^{2}+1 \right ) ^{-1/2\,{\frac{k}{b\mu }}} \]

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112.3 Problem 3

problem number 924

Added March 9, 2019.

Problem Chapter 4.6.4.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ x w_x + y w_y = a x \cot (\lambda x+ \mu y) w \]

Mathematica

\[ \left \{\left \{w(x,y)\to c_1\left (\frac{y}{x}\right ) \sin ^{\frac{a}{\lambda +\frac{\mu y}{x}}}\left (x \left (\lambda +\frac{\mu y}{x}\right )\right )\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({\frac{y}{x}} \right ) \left ( \left ( \cot \left ( \lambda \,x+\mu \,y \right ) \right ) ^{2}+1 \right ) ^{-1/2\,{a \left ({\frac{\mu \,y}{x}}+\lambda \right ) ^{-1}}} \]

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112.4 Problem 4

problem number 925

Added March 9, 2019.

Problem Chapter 4.6.4.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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112.5 Problem 5

problem number 926

Added March 9, 2019.

Problem Chapter 4.6.4.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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