82 HFOPDE, chapter 3.6.1

 82.1 Problem 1
 82.2 Problem 2
 82.3 Problem 3
 82.4 Problem 4
 82.5 Problem 5

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82.1 Problem 1

problem number 729

Added Feb. 11, 2019.

Problem Chapter 3.6.1.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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82.2 Problem 2

problem number 730

Added Feb. 11, 2019.

Problem Chapter 3.6.1.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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82.3 Problem 3

problem number 731

Added Feb. 11, 2019.

Problem Chapter 3.6.1.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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82.4 Problem 4

problem number 732

Added Feb. 11, 2019.

Problem Chapter 3.6.1.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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82.5 Problem 5

problem number 733

Added Feb. 11, 2019.

Problem Chapter 3.6.1.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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