74 HFOPDE, chapter 3.4.1

 74.1 Problem 1
 74.2 Problem 2
 74.3 Problem 3
 74.4 Problem 4
 74.5 Problem 5

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74.1 Problem 1

problem number 692

Added Feb. 9, 2019.

Problem Chapter 3.4.1.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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74.2 Problem 2

problem number 693

Added Feb. 9, 2019.

Problem Chapter 3.4.1.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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74.3 Problem 3

problem number 694

Added Feb. 9, 2019.

Problem Chapter 3.4.1.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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74.4 Problem 4

problem number 695

Added Feb. 9, 2019.

Problem Chapter 3.4.1.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

\[ \text{Kernel exception} \] Kernel Exception

Maple

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

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74.5 Problem 5

problem number 696

Added Feb. 9, 2019.

Problem Chapter 3.4.1.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

\[ w \left ( x,y \right ) =\int ^{x}\!{\frac{c \left ( \sinh \left ( \mu \,{\it \_a} \right ) \right ) ^{m}}{a}}+{\frac{s}{a} \left ( \sinh \left ({\frac{\beta }{\lambda }\ln \left ( -\tanh \left ( 1/2\,{\frac{\lambda \,b}{a} \left ( -{\frac{x\lambda \,b+2\,\arctanh \left ({{\rm e}^{\lambda \,y}} \right ) a}{\lambda \,b}}+{\it \_a} \right ) } \right ) \right ) } \right ) \right ) ^{k}}{d{\it \_a}}+{\it \_F1} \left ( -{\frac{x\lambda \,b+2\,\arctanh \left ({{\rm e}^{\lambda \,y}} \right ) a}{\lambda \,b}} \right ) \]