75 HFOPDE, chapter 3.4.2

 75.1 Problem 1
 75.2 Problem 2
 75.3 Problem 3
 75.4 Problem 4
 75.5 Problem 5

_______________________________________________________________________________________

75.1 Problem 1

problem number 697

Added Feb. 9, 2019.

Problem Chapter 3.4.2.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

_______________________________________________________________________________________

75.2 Problem 2

problem number 698

Added Feb. 9, 2019.

Problem Chapter 3.4.2.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

_______________________________________________________________________________________

75.3 Problem 3

problem number 699

Added Feb. 9, 2019.

Problem Chapter 3.4.2.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

_______________________________________________________________________________________

75.4 Problem 4

problem number 700

Added Feb. 9, 2019.

Problem Chapter 3.4.2.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

\[ \text{Timed out} \]

Maple

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

_______________________________________________________________________________________

75.5 Problem 5

problem number 701

Added Feb. 9, 2019.

Problem Chapter 3.4.2.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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