102 HFOPDE, chapter 4.4.1

 102.1 Problem 1
 102.2 Problem 2
 102.3 Problem 3
 102.4 Problem 4
 102.5 Problem 5

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102.1 Problem 1

problem number 869

Added Feb. 23, 2019.

Problem Chapter 4.4.1.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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102.2 Problem 2

problem number 870

Added Feb. 23, 2019.

Problem Chapter 4.4.1.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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102.3 Problem 3

problem number 871

Added Feb. 23, 2019.

Problem Chapter 4.4.1.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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102.4 Problem 4

problem number 872

Added Feb. 23, 2019.

Problem Chapter 4.4.1.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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102.5 Problem 5

problem number 873

Added Feb. 23, 2019.

Problem Chapter 4.4.1.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

\[ \left \{\left \{w(x,y)\to c_1() \exp \left (\frac{s x \sinh ^k(\beta y)}{a}-\frac{c \cosh (\mu x) \sinh ^{m+1}(\mu x) \left (-\sinh ^2(\mu x)\right )^{-\frac{m}{2}-\frac{1}{2}} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{1-m}{2},\frac{3}{2},\cosh ^2(\mu x)\right )}{a \mu }\right )\right \}\right \} \]

Maple

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