107 HFOPDE, chapter 4.4.3

 107.1 Problem 1
 107.2 Problem 2
 107.3 Problem 3
 107.4 Problem 4
 107.5 Problem 5

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107.1 Problem 1

problem number 895

Added Feb. 23, 2019.

Problem Chapter 4.4.3.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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107.2 Problem 2

problem number 896

Added Feb. 23, 2019.

Problem Chapter 4.4.3.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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107.3 Problem 3

problem number 897

Added Feb. 23, 2019.

Problem Chapter 4.4.3.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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107.4 Problem 4

problem number 898

Added Feb. 23, 2019.

Problem Chapter 4.4.3.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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107.5 Problem 5

problem number 899

Added Feb. 23, 2019.

Problem Chapter 4.4.3.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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