103 HFOPDE, chapter 4.4.2

 103.1 Problem 1
 103.2 Problem 2
 103.3 Problem 3
 103.4 Problem 4
 103.5 Problem 5

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103.1 Problem 1

problem number 874

Added Feb. 23, 2019.

Problem Chapter 4.4.2.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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103.2 Problem 2

problem number 875

Added Feb. 23, 2019.

Problem Chapter 4.4.2.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({\frac{ay-bx}{a}} \right ){{\rm e}^{{\frac{c}{\lambda \,a+\mu \,b}\sinh \left ({\frac{\mu \, \left ( ay-bx \right ) +ax\lambda +\mu \,bx}{a}} \right ) }}} \]

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103.3 Problem 3

problem number 876

Added Feb. 23, 2019.

Problem Chapter 4.4.2.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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103.4 Problem 4

problem number 877

Added Feb. 23, 2019.

Problem Chapter 4.4.2.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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103.5 Problem 5

problem number 878

Added Feb. 23, 2019.

Problem Chapter 4.4.2.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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