90 HFOPDE, chapter 3.7.4

 90.1 Problem 1
 90.2 Problem 2
 90.3 Problem 3
 90.4 Problem 4
 90.5 Problem 5

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90.1 Problem 1

problem number 770

Added Feb. 11, 2019.

Problem Chapter 3.7.4.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b w_y = c \arccot \frac{x}{\lambda }+ k \arccot \frac{y}{\beta } \]

Mathematica

\[ \left \{\left \{w(x,y)\to \frac{a \beta k \log \left (a^2 \beta ^2+(a y-b x)^2+2 b x (a y-b x)+b^2 x^2\right )+2 a b c_1\left (\frac{a y-b x}{a}\right )-2 a k y \tan ^{-1}\left (\frac{y}{\beta }\right )+2 b k x \tan ^{-1}\left (\frac{y}{\beta }\right )+2 b k x \cot ^{-1}\left (\frac{y}{\beta }\right )+b c \lambda \log \left (\lambda ^2+x^2\right )+2 b c x \cot ^{-1}\left (\frac{x}{\lambda }\right )}{2 a b}\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) =1/2\,{\frac{x}{ab} \left ( \pi \,bc+\pi \,bk-2\,\arctan \left ({\frac{x}{\lambda }} \right ) bc \right ) }-{\frac{ky}{b}\arctan \left ({\frac{bx}{a\beta }}+{\frac{ay-bx}{a\beta }} \right ) }+1/2\,{\frac{1}{ab} \left ( c\lambda \,\ln \left ({\frac{{x}^{2}}{{\lambda }^{2}}}+1 \right ) b+\beta \,k\ln \left ( \left ({\frac{bx}{a\beta }}+{\frac{ay-bx}{a\beta }} \right ) ^{2}+1 \right ) a+2\,{\it \_F1} \left ({\frac{ay-bx}{a}} \right ) ba \right ) } \]

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90.2 Problem 2

problem number 771

Added Feb. 11, 2019.

Problem Chapter 3.7.4.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b w_y = c \arccot (\lambda x+\beta y) \]

Mathematica

\[ \left \{\left \{w(x,y)\to \frac{c \left (a \log \left (a^2 \left (\beta ^2 y^2+2 \beta \lambda x y+\lambda ^2 x^2+1\right )\right )+2 \beta (b x-a y) \tan ^{-1}(\beta y+\lambda x)+2 x (a \lambda +b \beta ) \cot ^{-1}(\beta y+\lambda x)\right )}{2 a (a \lambda +b \beta )}+c_1\left (y-\frac{b x}{a}\right )\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) =1/2\,{\frac{1}{ \left ( \lambda \,a+b\beta \right ) a} \left ( c\ln \left ({\beta }^{2}{y}^{2}+2\,\beta \,\lambda \,xy+{\lambda }^{2}{x}^{2}+1 \right ) a+ \left ( 2\,{a}^{2}\lambda +2\,ab\beta \right ){\it \_F1} \left ({\frac{ay-bx}{a}} \right ) -2\,c \left ( a \left ( \beta \,y+\lambda \,x \right ) \arctan \left ( \beta \,y+\lambda \,x \right ) -1/2\,x\pi \, \left ( \lambda \,a+b\beta \right ) \right ) \right ) } \]

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90.3 Problem 3

problem number 772

Added Feb. 11, 2019.

Problem Chapter 3.7.4.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ x w_x + y w_y = a x \arccot (\lambda x+\beta y) \]

Mathematica

\[ \left \{\left \{w(x,y)\to a x \left (\frac{\log \left (\beta ^2 y^2+2 \beta \lambda x y+\lambda ^2 x^2+1\right )}{2 \beta y+2 \lambda x}+\cot ^{-1}(\beta y+\lambda x)\right )+c_1\left (\frac{y}{x}\right )\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={\frac{1}{2\,\beta \,y+2\,\lambda \,x} \left ( \ln \left ({\beta }^{2}{y}^{2}+2\,\beta \,\lambda \,xy+{\lambda }^{2}{x}^{2}+1 \right ) ax+ \left ( \beta \,y+\lambda \,x \right ) \left ( a\pi \,x-2\,ax\arctan \left ( \beta \,y+\lambda \,x \right ) +2\,{\it \_F1} \left ({\frac{y}{x}} \right ) \right ) \right ) } \]

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90.4 Problem 4

problem number 773

Added Feb. 11, 2019.

Problem Chapter 3.7.4.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

\[ w \left ( x,y \right ) =\int ^{x}\! \left ( \pi /2-\arctan \left ( \mu \,{\it \_a} \right ) \right ) ^{m}+{\frac{1}{a} \left ( \pi /2+\arctan \left ( -1/2\,{\frac{b\beta \,{\it \_a}\,\pi }{a}}+{\frac{b\beta \,{\it \_a}\,\arctan \left ( \lambda \,{\it \_a} \right ) }{a}}+1/2\,{\frac{ \left ( x\lambda \,b\pi -2\,bx\arctan \left ( \lambda \,x \right ) \lambda -2\,\lambda \,ya+b\ln \left ({\lambda }^{2}{x}^{2}+1 \right ) \right ) \beta }{\lambda \,a}}-1/2\,{\frac{b\beta \,\ln \left ({{\it \_a}}^{2}{\lambda }^{2}+1 \right ) }{\lambda \,a}} \right ) \right ) ^{k}}{d{\it \_a}}+{\it \_F1} \left ( -1/2\,{\frac{x\lambda \,b\pi -2\,bx\arctan \left ( \lambda \,x \right ) \lambda -2\,\lambda \,ya+b\ln \left ({\lambda }^{2}{x}^{2}+1 \right ) }{\lambda \,a}} \right ) \]

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90.5 Problem 5

problem number 774

Added Feb. 11, 2019.

Problem Chapter 3.7.4.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

\[ w \left ( x,y \right ) =\int ^{y}\!{\frac{a}{b{\rm arccot} \left (\lambda \,{\it \_b}\right )} \left ( \pi /2-\arctan \left ( 2\,{\frac{\mu \,a\int \! \left ( \pi -2\,\arctan \left ( \lambda \,{\it \_b} \right ) \right ) ^{-1}\,{\rm d}{\it \_b}}{b}}+\mu \, \left ( -\int \!2\,{\frac{a}{b \left ( \pi -2\,\arctan \left ( \lambda \,y \right ) \right ) }}\,{\rm d}y+x \right ) \right ) \right ) ^{m}}+{\frac{ \left ( \pi /2-\arctan \left ( \beta \,{\it \_b} \right ) \right ) ^{k}}{b{\rm arccot} \left (\lambda \,{\it \_b}\right )}}{d{\it \_b}}+{\it \_F1} \left ( -\int \!2\,{\frac{a}{b \left ( \pi -2\,\arctan \left ( \lambda \,y \right ) \right ) }}\,{\rm d}y+x \right ) \]