6.4.21 7.2

   6.4.21.1 [1146] Problem 1
   6.4.21.2 [1147] Problem 2
   6.4.21.3 [1148] Problem 3
   6.4.21.4 [1149] Problem 4
   6.4.21.5 [1150] Problem 5

6.4.21.1 [1146] Problem 1

problem number 1146

Added March 9, 2019.

Problem Chapter 4.7.2.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

\[w \left ( x,y \right ) ={\it \_F1} \left ({\frac{ya-bx}{a}} \right ){{\rm e}^{{\frac{1}{ab} \left ( a\arccos \left ({\frac{y}{\beta }} \right ) ky-\sqrt{{\frac{{\beta }^{2}-{y}^{2}}{{\beta }^{2}}}}a\beta \,k-\sqrt{-{\frac{{x}^{2}}{{\lambda }^{2}}}+1}bc\lambda +\arccos \left ({\frac{x}{\lambda }} \right ) bcx \right ) }}}\]

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6.4.21.2 [1147] Problem 2

problem number 1147

Added March 9, 2019.

Problem Chapter 4.7.2.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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6.4.21.3 [1148] Problem 3

problem number 1148

Added March 9, 2019.

Problem Chapter 4.7.2.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

\[\left \{\left \{w(x,y)\to c_1\left (y-\frac{b x}{a}\right ) \exp \left (\frac{\left (a^2+2 \beta ^2 (b x-a y)^2\right ) \sin ^{-1}(\beta y+\lambda x)-a \sqrt{-\beta ^2 y^2-2 \beta \lambda x y-\lambda ^2 x^2+1} (-3 a \beta y+a \lambda x+4 b \beta x)+2 x^2 (a \lambda +b \beta )^2 \cos ^{-1}(\beta y+\lambda x)}{4 (a \lambda +b \beta )^2}\right )\right \}\right \}\]

Maple

\[w \left ( x,y \right ) ={\it \_F1} \left ({\frac{ya-bx}{a}} \right ){{\rm e}^{1/2\,{\frac{ \left ( \left ( \left ( -1/2\,\lambda \,x+3/2\,\beta \,y \right ) a-2\,bx\beta \right ) \sqrt{-{\beta }^{2}{y}^{2}-2\,\beta \,\lambda \,xy-{\lambda }^{2}{x}^{2}+1}+ \left ( \beta \,y+\lambda \,x \right ) \left ( \left ( -\beta \,y+\lambda \,x \right ) a+2\,bx\beta \right ) \arccos \left ( \beta \,y+\lambda \,x \right ) +1/2\,\arcsin \left ( \beta \,y+\lambda \,x \right ) a \right ) a}{ \left ( a\lambda +b\beta \right ) ^{2}}}}}\]

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6.4.21.4 [1149] Problem 4

problem number 1149

Added March 9, 2019.

Problem Chapter 4.7.2.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

\[w \left ( x,y \right ) ={\it \_F1} \left ({\frac{1}{ \left ( n+2 \right ) a\lambda } \left ( b{2}^{-n}{2}^{n} \left ({\frac{ \left ( n+2 \right ) \LommelS 1 \left ( n+1/2,1/2,\arccos \left ( \lambda \,x \right ) \right ) }{\sqrt{\arccos \left ( \lambda \,x \right ) }}}-\sqrt{\arccos \left ( \lambda \,x \right ) }\LommelS 1 \left ( n+3/2,3/2,\arccos \left ( \lambda \,x \right ) \right ) + \left ( \arccos \left ( \lambda \,x \right ) \right ) ^{n+1} \right ) \sqrt{-{\lambda }^{2}{x}^{2}+1}+\lambda \, \left ( n+2 \right ) \left ( -\sqrt{\arccos \left ( \lambda \,x \right ) }bx{2}^{n}{2}^{-n}\LommelS 1 \left ( n+1/2,1/2,\arccos \left ( \lambda \,x \right ) \right ) +ya \right ) \right ) } \right ){{\rm e}^{\int ^{x}\!{\frac{1}{a} \left ( c \left ( \arccos \left ( \mu \,{\it \_b} \right ) \right ) ^{m}+s \left ( \pi -\arccos \left ({\frac{\beta }{ \left ( n+2 \right ) a\lambda } \left ( b{2}^{n} \left ({\frac{ \left ( n+2 \right ) \LommelS 1 \left ( n+1/2,1/2,\arccos \left ({\it \_b}\,\lambda \right ) \right ) }{\sqrt{\arccos \left ({\it \_b}\,\lambda \right ) }}}-\sqrt{\arccos \left ({\it \_b}\,\lambda \right ) }\LommelS 1 \left ( n+3/2,3/2,\arccos \left ({\it \_b}\,\lambda \right ) \right ) + \left ( \arccos \left ({\it \_b}\,\lambda \right ) \right ) ^{n+1} \right ){2}^{-n}\sqrt{-{{\it \_b}}^{2}{\lambda }^{2}+1}+\lambda \, \left ( n+2 \right ) \left ( -{2}^{-n}\sqrt{\arccos \left ({\it \_b}\,\lambda \right ) }b\LommelS 1 \left ( n+1/2,1/2,\arccos \left ({\it \_b}\,\lambda \right ) \right ){\it \_b}\,{2}^{n}+a\int \!{\frac{b \left ( \arccos \left ( \lambda \,x \right ) \right ) ^{n}}{a}}\,{\rm d}x-ya \right ) \right ) } \right ) \right ) ^{k} \right ) }{d{\it \_b}}}}\]

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6.4.21.5 [1150] Problem 5

problem number 1150

Added March 9, 2019.

Problem Chapter 4.7.2.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

\[w \left ( x,y \right ) ={\it \_F1} \left ({\frac{1}{ \left ( n-2 \right ) \lambda \,b} \left ({2}^{-n}{2}^{n} \left ({\frac{ \left ( n-2 \right ) \LommelS 1 \left ( -n+1/2,1/2,\arccos \left ( y\lambda \right ) \right ) }{\sqrt{\arccos \left ( y\lambda \right ) }}}+\sqrt{\arccos \left ( y\lambda \right ) }\LommelS 1 \left ( -n+3/2,3/2,\arccos \left ( y\lambda \right ) \right ) - \left ( \arccos \left ( y\lambda \right ) \right ) ^{-n+1} \right ) a\sqrt{-{y}^{2}{\lambda }^{2}+1}-\lambda \, \left ( n-2 \right ) \left ( a\sqrt{\arccos \left ( y\lambda \right ) }\LommelS 1 \left ( -n+1/2,1/2,\arccos \left ( y\lambda \right ) \right ) y{2}^{n}{2}^{-n}-bx \right ) \right ) } \right ){{\rm e}^{\int ^{y}\!{\frac{ \left ( \arccos \left ({\it \_b}\,\lambda \right ) \right ) ^{-n}}{b} \left ( c \left ( \arccos \left ({\frac{\mu }{ \left ( n-2 \right ) \lambda \,b} \left ( -{2}^{n} \left ({\frac{ \left ( n-2 \right ) \LommelS 1 \left ( -n+1/2,1/2,\arccos \left ({\it \_b}\,\lambda \right ) \right ) }{\sqrt{\arccos \left ({\it \_b}\,\lambda \right ) }}}+\sqrt{\arccos \left ({\it \_b}\,\lambda \right ) }\LommelS 1 \left ( -n+3/2,3/2,\arccos \left ({\it \_b}\,\lambda \right ) \right ) - \left ( \arccos \left ({\it \_b}\,\lambda \right ) \right ) ^{-n+1} \right ) a{2}^{-n}\sqrt{-{{\it \_b}}^{2}{\lambda }^{2}+1}+\lambda \, \left ( n-2 \right ) \left ( a{2}^{-n}\sqrt{\arccos \left ({\it \_b}\,\lambda \right ) }\LommelS 1 \left ( -n+1/2,1/2,\arccos \left ({\it \_b}\,\lambda \right ) \right ){\it \_b}\,{2}^{n}-a\int \! \left ( \arccos \left ( y\lambda \right ) \right ) ^{-n}\,{\rm d}y+bx \right ) \right ) } \right ) \right ) ^{m}+s \left ( \arccos \left ( \beta \,{\it \_b} \right ) \right ) ^{k} \right ) }{d{\it \_b}}}}\]

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