145 HFOPDE, chapter 5.7.3

 145.1 Problem 1
 145.2 Problem 2
 145.3 Problem 3
 145.4 Problem 4
 145.5 Problem 5

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145.1 Problem 1

problem number 1142

Added April 13, 2019.

Problem Chapter 5.7.3.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b w_y = w + c_1 \arctan ^k(\lambda x) + c_2 \arctan ^n(\beta y) \]

Mathematica

\[ \text{\$Aborted} \]

Maple

\[ w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac{1}{a}{{\rm e}^{-{\frac{{\it \_a}}{a}}}} \left ({\it c2}\, \left ( \arctan \left ({\frac{\beta \, \left ( ya-b \left ( x-{\it \_a} \right ) \right ) }{a}} \right ) \right ) ^{n}+{\it c1}\, \left ( \arctan \left ( \lambda \,{\it \_a} \right ) \right ) ^{k} \right ) }{d{\it \_a}}+{\it \_F1} \left ({\frac{ya-bx}{a}} \right ) \right ){{\rm e}^{{\frac{x}{a}}}} \]

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145.2 Problem 2

problem number 1143

Added April 13, 2019.

Problem Chapter 5.7.3.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

\[ \left \{\left \{w(x,y)\to e^{\frac{c x}{a}} \left (\int _1^x \frac{e^{-\frac{c K[1]}{a}} \tan ^{-1}(\lambda K[1])^k \tan ^{-1}\left (\beta \left (\frac{b (K[1]-x)}{a}+y\right )\right )^n}{a} \, dK[1]+c_1\left (y-\frac{b x}{a}\right )\right )\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac{ \left ( \arctan \left ( \lambda \,{\it \_a} \right ) \right ) ^{k}}{a} \left ( \arctan \left ({\frac{\beta \, \left ( ya-b \left ( x-{\it \_a} \right ) \right ) }{a}} \right ) \right ) ^{n}{{\rm e}^{-{\frac{{\it \_a}\,c}{a}}}}}{d{\it \_a}}+{\it \_F1} \left ({\frac{ya-bx}{a}} \right ) \right ){{\rm e}^{{\frac{cx}{a}}}} \]

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145.3 Problem 3

problem number 1144

Added April 13, 2019.

Problem Chapter 5.7.3.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b w_y = \left ( c_1 \arctan (\lambda _1 x) + c_2 \arctan (\lambda _2 y)\right ) w+ s_1 \arctan ^n(\beta _1 x)+ s_2 \arctan ^k(\beta _2 y) \]

Mathematica

\[ \text{\$Aborted} \]

Maple

\[ w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac{1}{a}{{\rm e}^{{\frac{1}{ab} \left ( - \left ( \left ({\it \_a}-x \right ) b+ya \right ){\it c2}\,\arctan \left ({\frac{\lambda 2\, \left ( ya-b \left ( x-{\it \_a} \right ) \right ) }{a}} \right ) -{\it c1}\,{\it \_a}\,\arctan \left ( \lambda 1\,{\it \_a} \right ) b \right ) }}} \left ({{\it \_a}}^{2}{\lambda 1}^{2}+1 \right ) ^{1/2\,{\frac{{\it c1}}{a\lambda 1}}} \left ({\frac{ \left ( ya-b \left ( x-{\it \_a} \right ) \right ) ^{2}{\lambda 2}^{2}+{a}^{2}}{{a}^{2}}} \right ) ^{1/2\,{\frac{{\it c2}}{b\lambda 2}}} \left ({\it s2}\, \left ( \arctan \left ({\frac{\beta 2\, \left ( ya-b \left ( x-{\it \_a} \right ) \right ) }{a}} \right ) \right ) ^{k}+{\it s1}\, \left ( \arctan \left ( \beta 1\,{\it \_a} \right ) \right ) ^{n} \right ) }{d{\it \_a}}+{\it \_F1} \left ({\frac{ya-bx}{a}} \right ) \right ) \left ({\lambda 1}^{2}{x}^{2}+1 \right ) ^{-1/2\,{\frac{{\it c1}}{a\lambda 1}}} \left ({\lambda 2}^{2}{y}^{2}+1 \right ) ^{-1/2\,{\frac{{\it c2}}{b\lambda 2}}}{{\rm e}^{{\frac{a\arctan \left ( \lambda 2\,y \right ) y{\it c2}+{\it c1}\,x\arctan \left ( \lambda 1\,x \right ) b}{ab}}}} \]

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145.4 Problem 4

problem number 1145

Added April 13, 2019.

Problem Chapter 5.7.3.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b \arctan ^m(\mu x) w_y = c \arctan ^k(\nu x) w + p \arctan ^n(\beta y) \]

Mathematica

\[ \text{\$Aborted} \]

Maple

\[ w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac{p}{a} \left ( \arctan \left ({\frac{\beta }{a} \left ( b\int \! \left ( \arctan \left ({\it \_f}\,\mu \right ) \right ) ^{m}\,{\rm d}{\it \_f}+ \left ( -\int \!{\frac{b \left ( \arctan \left ( \mu \,x \right ) \right ) ^{m}}{a}}\,{\rm d}x+y \right ) a \right ) } \right ) \right ) ^{n}{{\rm e}^{-{\frac{c\int \! \left ( \arctan \left ( \nu \,{\it \_f} \right ) \right ) ^{k}\,{\rm d}{\it \_f}}{a}}}}}{d{\it \_f}}+{\it \_F1} \left ( -\int \!{\frac{b \left ( \arctan \left ( \mu \,x \right ) \right ) ^{m}}{a}}\,{\rm d}x+y \right ) \right ){{\rm e}^{\int \!{\frac{ \left ( \arctan \left ( \nu \,x \right ) \right ) ^{k}c}{a}}\,{\rm d}x}} \]

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145.5 Problem 5

problem number 1146

Added April 13, 2019.

Problem Chapter 5.7.3.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b \arctan ^m(\mu x) w_y = c \arctan ^k(\nu y) w + p \arctan ^n(\beta x) \]

Mathematica

\[ \text{\$Aborted} \]

Maple

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