6.4.12 4.5

   6.4.12.1 [1097] Problem 1
   6.4.12.2 [1098] Problem 2
   6.4.12.3 [1099] Problem 3
   6.4.12.4 [1100] Problem 4
   6.4.12.5 [1101] Problem 5
   6.4.12.6 [1102] Problem 6

6.4.12.1 [1097] Problem 1

problem number 1097

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

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

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6.4.12.2 [1098] Problem 2

problem number 1098

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

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

Maple

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

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6.4.12.3 [1099] Problem 3

problem number 1099

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

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

Maple

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

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6.4.12.4 [1100] Problem 4

problem number 1100

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

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

Maple

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

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6.4.12.5 [1101] Problem 5

problem number 1101

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

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

Maple

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

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6.4.12.6 [1102] Problem 6

problem number 1102

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

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

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