110 HFOPDE, chapter 4.6.2

 110.1 Problem 1
 110.2 Problem 2
 110.3 Problem 3
 110.4 Problem 4
 110.5 Problem 5

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110.1 Problem 1

problem number 912

Added March 9, 2019.

Problem Chapter 4.6.2.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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110.2 Problem 2

problem number 913

Added March 9, 2019.

Problem Chapter 4.6.2.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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110.3 Problem 3

problem number 914

Added March 9, 2019.

Problem Chapter 4.6.2.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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110.4 Problem 4

problem number 915

Added March 9, 2019.

Problem Chapter 4.6.2.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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110.5 Problem 5

problem number 916

Added March 9, 2019.

Problem Chapter 4.6.2.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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