106 HFOPDE, chapter 4.4.5

 106.1 Problem 1
 106.2 Problem 2
 106.3 Problem 3
 106.4 Problem 4
 106.5 Problem 5
 106.6 Problem 6

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106.1 Problem 1

problem number 889

Added Feb. 23, 2019.

Problem Chapter 4.4.5.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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106.2 Problem 2

problem number 890

Added Feb. 23, 2019.

Problem Chapter 4.4.5.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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106.3 Problem 3

problem number 891

Added Feb. 23, 2019.

Problem Chapter 4.4.5.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

\[ \left \{\left \{w(x,y)\to e^{\frac{b \sinh (\lambda x)}{\lambda }} c_1\left (\frac{\log \left (\tanh \left (\frac{\mu y}{2}\right )\right )-a \mu x}{\mu }\right )\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac{\mu \,xa+2\,\arctanh \left ({{\rm e}^{\mu \,y}} \right ) }{a\mu }} \right ){{\rm e}^{{\frac{b\sinh \left ( \lambda \,x \right ) }{\lambda }}}} \]

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106.4 Problem 4

problem number 892

Added Feb. 23, 2019.

Problem Chapter 4.4.5.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

\[ \left \{\left \{w(x,y)\to \cosh ^{\frac{b}{\lambda }}(\lambda x) c_1\left (\frac{\log \left (\tanh \left (\frac{\mu y}{2}\right )\right )-a \mu x}{\mu }\right )\right \}\right \} \]

Maple

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

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106.5 Problem 5

problem number 893

Added Feb. 23, 2019.

Problem Chapter 4.4.5.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

\[ \left \{\left \{w(x,y)\to \sqrt [a \lambda ]{\sinh \left (\frac{\lambda x}{2}\right )} \cosh ^{-\frac{1}{a \lambda }}\left (\frac{\lambda x}{2}\right ) c_1\left (\frac{2 a \lambda \tan ^{-1}\left (\tanh \left (\frac{\mu y}{2}\right )\right )-b \mu \log \left (\sinh \left (\frac{\lambda x}{2}\right )\right )+b \mu \log \left (\cosh \left (\frac{\lambda x}{2}\right )\right )}{a \lambda \mu }\right )\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({\frac{1}{\lambda } \left ( \int \!{\frac{ \left ( \cosh \left ( \mu \,y \right ) \right ) ^{-n}a}{b}}\,{\rm d}y\lambda +2\,\arctanh \left ({{\rm e}^{\lambda \,x}} \right ) \right ) } \right ){{\rm e}^{-2\,{\frac{\arctanh \left ({{\rm e}^{\lambda \,x}} \right ) }{\lambda \,a}}}} \]

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106.6 Problem 6

problem number 894

Added Feb. 23, 2019.

Problem Chapter 4.4.5.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

\[ \left \{\left \{w(x,y)\to \sqrt [a \lambda ]{\sinh (\lambda x)} c_1\left (-\frac{2 a \cosh (\mu y) \sinh ^{-\frac{b \mu }{a \lambda }}(\lambda x)}{\mu }\right )\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={\it \_C1}\, \left ({\frac{\sinh \left ( \lambda \,x \right ) +\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }} \right ) ^{-1/2\,{\frac{{\it \_c}_{{1}}}{\lambda }}} \left ({\frac{\sinh \left ( \lambda \,x \right ) -\cosh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }} \right ) ^{-1/2\,{\frac{{\it \_c}_{{1}}}{\lambda }}} \left ({\frac{\sinh \left ( \lambda \,x \right ) }{\cosh \left ( \lambda \,x \right ) }} \right ) ^{{\frac{{\it \_c}_{{1}}}{\lambda }}} \left ({\frac{\cosh \left ( \mu \,y \right ) +\sinh \left ( \mu \,y \right ) }{\sinh \left ( \mu \,y \right ) }} \right ) ^{1/2\,{\frac{a{\it \_c}_{{1}}-1}{\mu \,b}}} \left ({\frac{\cosh \left ( \mu \,y \right ) -\sinh \left ( \mu \,y \right ) }{\sinh \left ( \mu \,y \right ) }} \right ) ^{1/2\,{\frac{a{\it \_c}_{{1}}-1}{\mu \,b}}} \left ({\frac{\cosh \left ( \mu \,y \right ) }{\sinh \left ( \mu \,y \right ) }} \right ) ^{{\frac{-a{\it \_c}_{{1}}+1}{\mu \,b}}}{\it \_C2} \]