6.4.22 7.3

   6.4.22.1 [1151] Problem 1
   6.4.22.2 [1152] Problem 2
   6.4.22.3 [1153] Problem 3
   6.4.22.4 [1154] Problem 4
   6.4.22.5 [1155] Problem 5

6.4.22.1 [1151] Problem 1

problem number 1151

Added March 9, 2019.

Problem Chapter 4.7.3.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b w_y = \left ( c \arctan (\frac{x}{\lambda } + k \arctan (\frac{y}{\beta } ) \right ) w \]

Mathematica

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

Maple

\[w \left ( x,y \right ) ={\it \_F1} \left ({\frac{ya-bx}{a}} \right ) \left ({\frac{{x}^{2}}{{\lambda }^{2}}}+1 \right ) ^{-1/2\,{\frac{\lambda \,c}{a}}} \left ({\frac{{\beta }^{2}+{y}^{2}}{{\beta }^{2}}} \right ) ^{-1/2\,{\frac{\beta \,k}{b}}}{{\rm e}^{{\frac{1}{ab} \left ( a\arctan \left ({\frac{y}{\beta }} \right ) ky+cx\arctan \left ({\frac{x}{\lambda }} \right ) b \right ) }}}\]

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6.4.22.2 [1152] Problem 2

problem number 1152

Added March 9, 2019.

Problem Chapter 4.7.3.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

\[w \left ( x,y \right ) ={\it \_F1} \left ({\frac{ya-bx}{a}} \right ) \left ({\beta }^{2}{y}^{2}+2\,\beta \,\lambda \,xy+{\lambda }^{2}{x}^{2}+1 \right ) ^{-{\frac{c}{2\,a\lambda +2\,b\beta }}}{{\rm e}^{{\frac{c\arctan \left ( \beta \,y+\lambda \,x \right ) \left ( \beta \,y+\lambda \,x \right ) }{a\lambda +b\beta }}}}\]

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6.4.22.3 [1153] Problem 3

problem number 1153

Added March 9, 2019.

Problem Chapter 4.7.3.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

\[\left \{\left \{w(x,y)\to c_1\left (y-\frac{b x}{a}\right ) \exp \left (\frac{1}{4} \left (2 x^2 \tan ^{-1}(\beta y+\lambda x)+\frac{i (a+i \beta (b x-a y))^2 \log (a (\beta y+\lambda x+i))+i (b \beta x+a (-\beta y+i))^2 \log (-a (\beta y+\lambda x-i))-2 a x (a \lambda +b \beta )}{(a \lambda +b \beta )^2}\right )\right )\right \}\right \}\]

Maple

\[w \left ( x,y \right ) ={\it \_F1} \left ({\frac{ya-bx}{a}} \right ) \left ({\beta }^{2}{y}^{2}+2\,\beta \,\lambda \,xy+{\lambda }^{2}{x}^{2}+1 \right ) ^{1/2\,{\frac{ \left ( ya-bx \right ) a\beta }{ \left ( a\lambda +b\beta \right ) ^{2}}}}{{\rm e}^{1/2\,{\frac{ \left ( \left ( \left ( -{\beta }^{2}{y}^{2}+{\lambda }^{2}{x}^{2}+1 \right ) a+2\,b\beta \,x \left ( \beta \,y+\lambda \,x \right ) \right ) \arctan \left ( \beta \,y+\lambda \,x \right ) -a \left ( \beta \,y+\lambda \,x \right ) \right ) a}{ \left ( a\lambda +b\beta \right ) ^{2}}}}}\]

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6.4.22.4 [1154] Problem 4

problem number 1154

Added March 9, 2019.

Problem Chapter 4.7.3.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

\[\left \{\left \{w(x,y)\to c_1\left (y-\int _1^x\frac{b \tan ^{-1}(\lambda K[1])^n}{a}dK[1]\right ) \exp \left (\int _1^x\frac{s \tan ^{-1}\left (\beta \left (y-\int _1^x\frac{b \tan ^{-1}(\lambda K[1])^n}{a}dK[1]+\int _1^{K[2]}\frac{b \tan ^{-1}(\lambda K[1])^n}{a}dK[1]\right )\right ){}^k+c \tan ^{-1}(\mu K[2])^m}{a}dK[2]\right )\right \}\right \}\]

Maple

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

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6.4.22.5 [1155] Problem 5

problem number 1155

Added March 9, 2019.

Problem Chapter 4.7.3.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

\[\left \{\left \{w(x,y)\to c_1\left (\int _1^y\tan ^{-1}(\lambda K[1])^{-n}dK[1]-\frac{b x}{a}\right ) \exp \left (\int _1^y\frac{\tan ^{-1}(\lambda K[2])^{-n} \left (s \tan ^{-1}(\beta K[2])^k+c \tan ^{-1}\left (\frac{\mu \left (b x-a \int _1^y\tan ^{-1}(\lambda K[1])^{-n}dK[1]+a \int _1^{K[2]}\tan ^{-1}(\lambda K[1])^{-n}dK[1]\right )}{b}\right ){}^m\right )}{b}dK[2]\right )\right \}\right \}\]

Maple

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

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