85 HFOPDE, chapter 3.6.4

 85.1 Problem 1
 85.2 Problem 2
 85.3 Problem 3
 85.4 Problem 4
 85.5 Problem 5

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85.1 Problem 1

problem number 744

Added Feb. 11, 2019.

Problem Chapter 3.6.4.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

\[ w \left ( x,y \right ) =-1/2\,{\frac{1}{\mu \,b\lambda \,a} \left ( -2\,{\it \_F1} \left ({\frac{ay-bx}{a}} \right ) \mu \,b\lambda \,a+k\ln \left ( \left ( \cot \left ( \mu \,y \right ) \right ) ^{2}+1 \right ) \lambda \,a+c\ln \left ( \left ( \cot \left ( \lambda \,x \right ) \right ) ^{2}+1 \right ) \mu \,b \right ) } \]

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85.2 Problem 2

problem number 745

Added Feb. 11, 2019.

Problem Chapter 3.6.4.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

\[ w \left ( x,y \right ) =-1/2\,{\frac{c}{\lambda \,a+\mu \,b}\ln \left ( \left ( \cot \left ({\frac{\mu \, \left ( ay-bx \right ) +ax\lambda +\mu \,bx}{a}} \right ) \right ) ^{2}+1 \right ) }+{\it \_F1} \left ({\frac{ay-bx}{a}} \right ) \]

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85.3 Problem 3

problem number 746

Added Feb. 11, 2019.

Problem Chapter 3.6.4.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

\[ w \left ( x,y \right ) =-1/2\,{a\ln \left ( \left ( \cot \left ( x \left ({\frac{\mu \,y}{x}}+\lambda \right ) \right ) \right ) ^{2}+1 \right ) \left ({\frac{\mu \,y}{x}}+\lambda \right ) ^{-1}}+{\it \_F1} \left ({\frac{y}{x}} \right ) \]

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85.4 Problem 4

problem number 747

Added Feb. 11, 2019.

Problem Chapter 3.6.4.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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85.5 Problem 5

problem number 748

Added Feb. 11, 2019.

Problem Chapter 3.6.4.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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