78 HFOPDE, chapter 3.4.5

 78.1 Problem 1
 78.2 Problem 2
 78.3 Problem 3
 78.4 Problem 4
 78.5 Problem 5

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78.1 Problem 1

problem number 712

Added Feb. 11, 2019.

Problem Chapter 3.4.5.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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78.2 Problem 2

problem number 713

Added Feb. 11, 2019.

Problem Chapter 3.4.5.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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78.3 Problem 3

problem number 714

Added Feb. 11, 2019.

Problem Chapter 3.4.5.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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78.4 Problem 4

problem number 715

Added Feb. 11, 2019.

Problem Chapter 3.4.5.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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78.5 Problem 5

problem number 716

Added Feb. 11, 2019.

Problem Chapter 3.4.5.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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