84 HFOPDE, chapter 3.6.3

 84.1 Problem 1
 84.2 Problem 2
 84.3 Problem 3
 84.4 Problem 4
 84.5 Problem 5

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84.1 Problem 1

problem number 739

Added Feb. 11, 2019.

Problem Chapter 3.6.3.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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84.2 Problem 2

problem number 740

Added Feb. 11, 2019.

Problem Chapter 3.6.3.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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84.3 Problem 3

problem number 741

Added Feb. 11, 2019.

Problem Chapter 3.6.3.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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84.4 Problem 4

problem number 742

Added Feb. 11, 2019.

Problem Chapter 3.6.3.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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84.5 Problem 5

problem number 743

Added Feb. 11, 2019.

Problem Chapter 3.6.3.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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