6.4.23 7.4

   6.4.23.1 [1156] Problem 1
   6.4.23.2 [1157] Problem 2
   6.4.23.3 [1158] Problem 3
   6.4.23.4 [1159] Problem 4
   6.4.23.5 [1160] Problem 5

6.4.23.1 [1156] Problem 1

problem number 1156

Added March 9, 2019.

Problem Chapter 4.7.4.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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6.4.23.2 [1157] Problem 2

problem number 1157

Added March 9, 2019.

Problem Chapter 4.7.4.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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6.4.23.3 [1158] Problem 3

problem number 1158

Added March 9, 2019.

Problem Chapter 4.7.4.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

\[w \left ( x,y \right ) ={\it \_F1} \left ({\frac{ya-bx}{a}} \right ) \left ({\beta }^{2}{y}^{2}+2\,\beta \,\lambda \,xy+{\lambda }^{2}{x}^{2}+1 \right ) ^{-1/2\,{\frac{ \left ( ya-bx \right ) a\beta }{ \left ( a\lambda +b\beta \right ) ^{2}}}}{{\rm e}^{1/4\,{\frac{-2\, \left ( \left ( -{\beta }^{2}{y}^{2}+{\lambda }^{2}{x}^{2}+1 \right ) a+2\,b\beta \,x \left ( \beta \,y+\lambda \,x \right ) \right ) a\arctan \left ( \beta \,y+\lambda \,x \right ) + \left ( \pi \,{\lambda }^{2}{x}^{2}+2\,\beta \,y+2\,\lambda \,x \right ){a}^{2}+2\,\pi \,ab\beta \,\lambda \,{x}^{2}+\pi \,{b}^{2}{\beta }^{2}{x}^{2}}{ \left ( a\lambda +b\beta \right ) ^{2}}}}}\]

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6.4.23.4 [1159] Problem 4

problem number 1159

Added March 9, 2019.

Problem Chapter 4.7.4.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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6.4.23.5 [1160] Problem 5

problem number 1160

Added March 9, 2019.

Problem Chapter 4.7.4.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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