77 HFOPDE, chapter 3.4.4

 77.1 Problem 1
 77.2 Problem 2
 77.3 Problem 3
 77.4 Problem 4
 77.5 Problem 5

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77.1 Problem 1

problem number 707

Added Feb. 11, 2019.

Problem Chapter 3.4.4.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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77.2 Problem 2

problem number 708

Added Feb. 11, 2019.

Problem Chapter 3.4.4.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

\[ w \left ( x,y \right ) =1/2\,{\frac{1}{\lambda \,a+\mu \,b} \left ( 2\,{\it \_F1} \left ({\frac{ay-bx}{a}} \right ) a\lambda +2\,{\it \_F1} \left ({\frac{ay-bx}{a}} \right ) b\mu -c\ln \left ({\rm coth} \left ({\frac{\mu \, \left ( ay-bx \right ) +ax\lambda +\mu \,bx}{a}}\right )-1 \right ) -c\ln \left ({\rm coth} \left ({\frac{\mu \, \left ( ay-bx \right ) +ax\lambda +\mu \,bx}{a}}\right )+1 \right ) \right ) } \]

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77.3 Problem 3

problem number 709

Added Feb. 11, 2019.

Problem Chapter 3.4.4.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

\[ w \left ( x,y \right ) =-1/2\,{\frac{1}{\lambda \,x+\mu \,y} \left ( \left ( \ln \left ({\rm coth} \left (x \left ({\frac{\mu \,y}{x}}+\lambda \right ) \right )-1 \right ) a+a\ln \left ({\rm coth} \left (x \left ({\frac{\mu \,y}{x}}+\lambda \right ) \right )+1 \right ) -2\,{\it \_F1} \left ({\frac{y}{x}} \right ) \lambda \right ) x-2\,{\it \_F1} \left ({\frac{y}{x}} \right ) \mu \,y \right ) } \]

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77.4 Problem 4

problem number 710

Added Feb. 11, 2019.

Problem Chapter 3.4.4.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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77.5 Problem 5

problem number 711

Added Feb. 11, 2019.

Problem Chapter 3.4.4.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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