109 HFOPDE, chapter 4.6.1

 109.1 Problem 1
 109.2 Problem 2
 109.3 Problem 3
 109.4 Problem 4
 109.5 Problem 5

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109.1 Problem 1

problem number 907

Added March 9, 2019.

Problem Chapter 4.6.1.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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109.2 Problem 2

problem number 908

Added March 9, 2019.

Problem Chapter 4.6.1.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

\[ \left \{\left \{w(x,y)\to c_1\left (y-\frac{b x}{a}\right ) e^{-\frac{c \cos (\lambda x)}{a \lambda }-\frac{k \cos (\mu y)}{b \mu }}\right \}\right \} \]

Maple

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

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109.3 Problem 3

problem number 909

Added March 9, 2019.

Problem Chapter 4.6.1.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

\[ \left \{\left \{w(x,y)\to c_1\left (\frac{y}{x}\right ) e^{-\frac{a x \cos (\lambda x+\mu y)}{\lambda x+\mu y}}\right \}\right \} \]

Maple

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

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109.4 Problem 4

problem number 910

Added March 9, 2019.

Problem Chapter 4.6.1.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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109.5 Problem 5

problem number 911

Added March 9, 2019.

Problem Chapter 4.6.1.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

\[ \left \{\left \{w(x,y)\to c_1() \exp \left (\frac{s x \sin ^k(\beta y)}{a}-\frac{c \cos (\mu x) \sin ^{m+1}(\mu x) \sin ^2(\mu x)^{-\frac{m}{2}-\frac{1}{2}} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{1-m}{2},\frac{3}{2},\cos ^2(\mu x)\right )}{a \mu }\right )\right \}\right \} \]

Maple

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