55 HFOPDE, chapter 2.7.2

 55.1 problem number 1
 55.2 problem number 2
 55.3 problem number 3
 55.4 problem number 4
 55.5 problem number 5
 55.6 problem number 6
 55.7 problem number 7
 55.8 problem number 8
 55.9 problem number 9
 55.10 problem number 10
 55.11 problem number 11
 55.12 problem number 12

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55.1 problem number 1

problem number 508

Added January 29, 2019.

Problem 2.7.2.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( a \arccos ^k(\lambda x) + b \right ) w_y = 0 \]

Mathematica

\[ \left \{\left \{w(x,y)\to c_1\left (-\frac{\left (\cos ^{-1}(\lambda x)^2\right )^{-k} \left (a \left (i \cos ^{-1}(\lambda x)\right )^k \cos ^{-1}(\lambda x)^k \text{Gamma}\left (k+1,-i \cos ^{-1}(\lambda x)\right )+a \left (-i \cos ^{-1}(\lambda x)\right )^k \cos ^{-1}(\lambda x)^k \text{Gamma}\left (k+1,i \cos ^{-1}(\lambda x)\right )+2 b \lambda x \left (\cos ^{-1}(\lambda x)^2\right )^k-2 \lambda y \left (\cos ^{-1}(\lambda x)^2\right )^k\right )}{2 \lambda }\right )\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -bx+{\frac{a{2}^{k}\sqrt{\pi }}{\lambda } \left ({\frac{ \left ( \arccos \left ( \lambda \,x \right ) \right ) ^{k+1}{2}^{-k}\sqrt{-{\lambda }^{2}{x}^{2}+1}}{\sqrt{\pi } \left ( k+2 \right ) }}-{\frac{{2}^{-k}\sqrt{\arccos \left ( \lambda \,x \right ) }\LommelS 1 \left ( k+3/2,3/2,\arccos \left ( \lambda \,x \right ) \right ) \sqrt{-{\lambda }^{2}{x}^{2}+1}}{\sqrt{\pi } \left ( k+2 \right ) }}-3\,{\frac{{2}^{-1-k} \left ( 2/3\,k+4/3 \right ) \left ( \lambda \,x\arccos \left ( \lambda \,x \right ) -\sqrt{-{\lambda }^{2}{x}^{2}+1} \right ) \LommelS 1 \left ( k+1/2,1/2,\arccos \left ( \lambda \,x \right ) \right ) }{\sqrt{\pi } \left ( k+2 \right ) \sqrt{\arccos \left ( \lambda \,x \right ) }}} \right ) }+y \right ) \]

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55.2 problem number 2

problem number 509

Added January 29, 2019.

Problem 2.7.2.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( a \arccos ^k(\lambda y) + b \right ) w_y = 0 \]

Mathematica

\[ \text{Timed out} \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -\int \! \left ( a \left ( \arccos \left ( \lambda \,y \right ) \right ) ^{k}+b \right ) ^{-1}\,{\rm d}y+x \right ) \]

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55.3 problem number 3

problem number 510

Added January 29, 2019.

Problem 2.7.2.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + k \arccos ^n(a x+b y+c) w_y = 0 \]

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -\int ^{{\frac{ax+by}{b}}}\! \left ( k \left ( \arccos \left ({\it \_a}\,b+c \right ) \right ) ^{n}b+a \right ) ^{-1}{d{\it \_a}}b+x \right ) \]

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55.4 problem number 4

problem number 511

Added January 29, 2019.

Problem 2.7.2.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + a \arccos ^k(\lambda x) \arccos ^n(\mu y) w_y = 0 \]

Mathematica

\[ \left \{\left \{w(x,y)\to c_1\left (\frac{\left (\cos ^{-1}(\lambda x)^2\right )^{-k} \left (-a \left (i \cos ^{-1}(\lambda x)\right )^k \cos ^{-1}(\lambda x)^k \text{Gamma}\left (k+1,-i \cos ^{-1}(\lambda x)\right )-a \left (-i \cos ^{-1}(\lambda x)\right )^k \cos ^{-1}(\lambda x)^k \text{Gamma}\left (k+1,i \cos ^{-1}(\lambda x)\right )+\frac{\lambda \left (\cos ^{-1}(\lambda x)^2\right )^k \cos ^{-1}(\mu y)^{-n} \left (\left (-i \cos ^{-1}(\mu y)\right )^n \text{Gamma}\left (1-n,-i \cos ^{-1}(\mu y)\right )+\left (i \cos ^{-1}(\mu y)\right )^n \text{Gamma}\left (1-n,i \cos ^{-1}(\mu y)\right )\right )}{\mu }\right )}{2 \lambda }\right )\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac{1}{ \left ( n-2 \right ) a\mu \,\sqrt{\arccos \left ( \mu \,y \right ) }} \left ( -\arccos \left ( \mu \,y \right ) y\LommelS 1 \left ( -n+1/2,1/2,\arccos \left ( \mu \,y \right ) \right ) \mu \,n+2\,\arccos \left ( \mu \,y \right ) y\LommelS 1 \left ( -n+1/2,1/2,\arccos \left ( \mu \,y \right ) \right ) \mu -{\frac{a{2}^{k}\sqrt{\pi }\mu \,n\sqrt{\arccos \left ( \mu \,y \right ) }}{\lambda } \left ({\frac{ \left ( \arccos \left ( \lambda \,x \right ) \right ) ^{k+1}{2}^{-k}\sqrt{-{\lambda }^{2}{x}^{2}+1}}{\sqrt{\pi } \left ( k+2 \right ) }}-{\frac{{2}^{-k}\sqrt{\arccos \left ( \lambda \,x \right ) }\LommelS 1 \left ( k+3/2,3/2,\arccos \left ( \lambda \,x \right ) \right ) \sqrt{-{\lambda }^{2}{x}^{2}+1}}{\sqrt{\pi } \left ( k+2 \right ) }}-3\,{\frac{{2}^{-1-k} \left ( 2/3\,k+4/3 \right ) \left ( \lambda \,x\arccos \left ( \lambda \,x \right ) -\sqrt{-{\lambda }^{2}{x}^{2}+1} \right ) \LommelS 1 \left ( k+1/2,1/2,\arccos \left ( \lambda \,x \right ) \right ) }{\sqrt{\pi } \left ( k+2 \right ) \sqrt{\arccos \left ( \lambda \,x \right ) }}} \right ) }+\arccos \left ( \mu \,y \right ) \LommelS 1 \left ( -n+3/2,3/2,\arccos \left ( \mu \,y \right ) \right ) \sqrt{-{\mu }^{2}{y}^{2}+1}+2\,{\frac{a{2}^{k}\sqrt{\pi }\mu \,\sqrt{\arccos \left ( \mu \,y \right ) }}{\lambda } \left ({\frac{ \left ( \arccos \left ( \lambda \,x \right ) \right ) ^{k+1}{2}^{-k}\sqrt{-{\lambda }^{2}{x}^{2}+1}}{\sqrt{\pi } \left ( k+2 \right ) }}-{\frac{{2}^{-k}\sqrt{\arccos \left ( \lambda \,x \right ) }\LommelS 1 \left ( k+3/2,3/2,\arccos \left ( \lambda \,x \right ) \right ) \sqrt{-{\lambda }^{2}{x}^{2}+1}}{\sqrt{\pi } \left ( k+2 \right ) }}-3\,{\frac{{2}^{-1-k} \left ( 2/3\,k+4/3 \right ) \left ( \lambda \,x\arccos \left ( \lambda \,x \right ) -\sqrt{-{\lambda }^{2}{x}^{2}+1} \right ) \LommelS 1 \left ( k+1/2,1/2,\arccos \left ( \lambda \,x \right ) \right ) }{\sqrt{\pi } \left ( k+2 \right ) \sqrt{\arccos \left ( \lambda \,x \right ) }}} \right ) }- \left ( \arccos \left ( \mu \,y \right ) \right ) ^{-n+3/2}\sqrt{-{\mu }^{2}{y}^{2}+1}+\sqrt{-{\mu }^{2}{y}^{2}+1}\LommelS 1 \left ( -n+1/2,1/2,\arccos \left ( \mu \,y \right ) \right ) n-2\,\sqrt{-{\mu }^{2}{y}^{2}+1}\LommelS 1 \left ( -n+1/2,1/2,\arccos \left ( \mu \,y \right ) \right ) \right ) } \right ) \]

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55.5 problem number 5

problem number 512

Added January 29, 2019.

Problem 2.7.2.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( y^2+\lambda (\arccos x)^n y- a^2 + a \lambda ( \arccos x)^n \right ) w_y = 0 \]

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac{1}{ \left ( 2+n \right ) \left ( a+y \right ) } \left ( \int \!{\frac{\sqrt{\arccos \left ( x \right ) }\LommelS 1 \left ( n-1/2,1/2,\arccos \left ( x \right ) \right ) x}{\sqrt{-{x}^{2}+1}}{{\rm e}^{-{\frac{-\LommelS 1 \left ( n+3/2,3/2,\arccos \left ( x \right ) \right ) \lambda \,\arccos \left ( x \right ) \sqrt{-{x}^{2}+1}-nx\lambda \,\arccos \left ( x \right ) \LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) +\lambda \,n\LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) \sqrt{-{x}^{2}+1}+\lambda \, \left ( \arccos \left ( x \right ) \right ) ^{n+3/2}\sqrt{-{x}^{2}+1}-2\,\lambda \,x\arccos \left ( x \right ) \LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) +2\,\lambda \,\LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) \sqrt{-{x}^{2}+1}+2\,ax\sqrt{\arccos \left ( x \right ) }n+4\,ax\sqrt{\arccos \left ( x \right ) }}{ \left ( 2+n \right ) \sqrt{\arccos \left ( x \right ) }}}}}}\,{\rm d}x\lambda \,{n}^{2}-\int \!{\frac{\LommelS 1 \left ( n-1/2,1/2,\arccos \left ( x \right ) \right ) }{\sqrt{\arccos \left ( x \right ) }}{{\rm e}^{-{\frac{-\LommelS 1 \left ( n+3/2,3/2,\arccos \left ( x \right ) \right ) \lambda \,\arccos \left ( x \right ) \sqrt{-{x}^{2}+1}-nx\lambda \,\arccos \left ( x \right ) \LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) +\lambda \,n\LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) \sqrt{-{x}^{2}+1}+\lambda \, \left ( \arccos \left ( x \right ) \right ) ^{n+3/2}\sqrt{-{x}^{2}+1}-2\,\lambda \,x\arccos \left ( x \right ) \LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) +2\,\lambda \,\LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) \sqrt{-{x}^{2}+1}+2\,ax\sqrt{\arccos \left ( x \right ) }n+4\,ax\sqrt{\arccos \left ( x \right ) }}{ \left ( 2+n \right ) \sqrt{\arccos \left ( x \right ) }}}}}}\,{\rm d}x\lambda \,{n}^{2}+\int \!{{\rm e}^{-{\frac{-\LommelS 1 \left ( n+3/2,3/2,\arccos \left ( x \right ) \right ) \lambda \,\arccos \left ( x \right ) \sqrt{-{x}^{2}+1}-nx\lambda \,\arccos \left ( x \right ) \LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) +\lambda \,n\LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) \sqrt{-{x}^{2}+1}+\lambda \, \left ( \arccos \left ( x \right ) \right ) ^{n+3/2}\sqrt{-{x}^{2}+1}-2\,\lambda \,x\arccos \left ( x \right ) \LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) +2\,\lambda \,\LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) \sqrt{-{x}^{2}+1}+2\,ax\sqrt{\arccos \left ( x \right ) }n+4\,ax\sqrt{\arccos \left ( x \right ) }}{ \left ( 2+n \right ) \sqrt{\arccos \left ( x \right ) }}}}}\,{\rm d}xyn+\int \!{{\rm e}^{-{\frac{-\LommelS 1 \left ( n+3/2,3/2,\arccos \left ( x \right ) \right ) \lambda \,\arccos \left ( x \right ) \sqrt{-{x}^{2}+1}-nx\lambda \,\arccos \left ( x \right ) \LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) +\lambda \,n\LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) \sqrt{-{x}^{2}+1}+\lambda \, \left ( \arccos \left ( x \right ) \right ) ^{n+3/2}\sqrt{-{x}^{2}+1}-2\,\lambda \,x\arccos \left ( x \right ) \LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) +2\,\lambda \,\LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) \sqrt{-{x}^{2}+1}+2\,ax\sqrt{\arccos \left ( x \right ) }n+4\,ax\sqrt{\arccos \left ( x \right ) }}{ \left ( 2+n \right ) \sqrt{\arccos \left ( x \right ) }}}}}\,{\rm d}xan+2\,\int \!{\frac{\sqrt{\arccos \left ( x \right ) }\LommelS 1 \left ( n-1/2,1/2,\arccos \left ( x \right ) \right ) x}{\sqrt{-{x}^{2}+1}}{{\rm e}^{-{\frac{-\LommelS 1 \left ( n+3/2,3/2,\arccos \left ( x \right ) \right ) \lambda \,\arccos \left ( x \right ) \sqrt{-{x}^{2}+1}-nx\lambda \,\arccos \left ( x \right ) \LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) +\lambda \,n\LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) \sqrt{-{x}^{2}+1}+\lambda \, \left ( \arccos \left ( x \right ) \right ) ^{n+3/2}\sqrt{-{x}^{2}+1}-2\,\lambda \,x\arccos \left ( x \right ) \LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) +2\,\lambda \,\LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) \sqrt{-{x}^{2}+1}+2\,ax\sqrt{\arccos \left ( x \right ) }n+4\,ax\sqrt{\arccos \left ( x \right ) }}{ \left ( 2+n \right ) \sqrt{\arccos \left ( x \right ) }}}}}}\,{\rm d}x\lambda \,n-2\,\int \!{\frac{\LommelS 1 \left ( n-1/2,1/2,\arccos \left ( x \right ) \right ) }{\sqrt{\arccos \left ( x \right ) }}{{\rm e}^{-{\frac{-\LommelS 1 \left ( n+3/2,3/2,\arccos \left ( x \right ) \right ) \lambda \,\arccos \left ( x \right ) \sqrt{-{x}^{2}+1}-nx\lambda \,\arccos \left ( x \right ) \LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) +\lambda \,n\LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) \sqrt{-{x}^{2}+1}+\lambda \, \left ( \arccos \left ( x \right ) \right ) ^{n+3/2}\sqrt{-{x}^{2}+1}-2\,\lambda \,x\arccos \left ( x \right ) \LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) +2\,\lambda \,\LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) \sqrt{-{x}^{2}+1}+2\,ax\sqrt{\arccos \left ( x \right ) }n+4\,ax\sqrt{\arccos \left ( x \right ) }}{ \left ( 2+n \right ) \sqrt{\arccos \left ( x \right ) }}}}}}\,{\rm d}x\lambda \,n+\int \!{\frac{\LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) }{ \left ( \arccos \left ( x \right ) \right ) ^{3/2}}{{\rm e}^{-{\frac{-\LommelS 1 \left ( n+3/2,3/2,\arccos \left ( x \right ) \right ) \lambda \,\arccos \left ( x \right ) \sqrt{-{x}^{2}+1}-nx\lambda \,\arccos \left ( x \right ) \LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) +\lambda \,n\LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) \sqrt{-{x}^{2}+1}+\lambda \, \left ( \arccos \left ( x \right ) \right ) ^{n+3/2}\sqrt{-{x}^{2}+1}-2\,\lambda \,x\arccos \left ( x \right ) \LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) +2\,\lambda \,\LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) \sqrt{-{x}^{2}+1}+2\,ax\sqrt{\arccos \left ( x \right ) }n+4\,ax\sqrt{\arccos \left ( x \right ) }}{ \left ( 2+n \right ) \sqrt{\arccos \left ( x \right ) }}}}}}\,{\rm d}x\lambda \,n-\int \!{\frac{x\LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) }{\sqrt{\arccos \left ( x \right ) }\sqrt{-{x}^{2}+1}}{{\rm e}^{-{\frac{-\LommelS 1 \left ( n+3/2,3/2,\arccos \left ( x \right ) \right ) \lambda \,\arccos \left ( x \right ) \sqrt{-{x}^{2}+1}-nx\lambda \,\arccos \left ( x \right ) \LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) +\lambda \,n\LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) \sqrt{-{x}^{2}+1}+\lambda \, \left ( \arccos \left ( x \right ) \right ) ^{n+3/2}\sqrt{-{x}^{2}+1}-2\,\lambda \,x\arccos \left ( x \right ) \LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) +2\,\lambda \,\LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) \sqrt{-{x}^{2}+1}+2\,ax\sqrt{\arccos \left ( x \right ) }n+4\,ax\sqrt{\arccos \left ( x \right ) }}{ \left ( 2+n \right ) \sqrt{\arccos \left ( x \right ) }}}}}}\,{\rm d}x\lambda \,n+2\,\int \!{{\rm e}^{-{\frac{-\LommelS 1 \left ( n+3/2,3/2,\arccos \left ( x \right ) \right ) \lambda \,\arccos \left ( x \right ) \sqrt{-{x}^{2}+1}-nx\lambda \,\arccos \left ( x \right ) \LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) +\lambda \,n\LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) \sqrt{-{x}^{2}+1}+\lambda \, \left ( \arccos \left ( x \right ) \right ) ^{n+3/2}\sqrt{-{x}^{2}+1}-2\,\lambda \,x\arccos \left ( x \right ) \LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) +2\,\lambda \,\LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) \sqrt{-{x}^{2}+1}+2\,ax\sqrt{\arccos \left ( x \right ) }n+4\,ax\sqrt{\arccos \left ( x \right ) }}{ \left ( 2+n \right ) \sqrt{\arccos \left ( x \right ) }}}}}\,{\rm d}xy+2\,\int \!{{\rm e}^{-{\frac{-\LommelS 1 \left ( n+3/2,3/2,\arccos \left ( x \right ) \right ) \lambda \,\arccos \left ( x \right ) \sqrt{-{x}^{2}+1}-nx\lambda \,\arccos \left ( x \right ) \LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) +\lambda \,n\LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) \sqrt{-{x}^{2}+1}+\lambda \, \left ( \arccos \left ( x \right ) \right ) ^{n+3/2}\sqrt{-{x}^{2}+1}-2\,\lambda \,x\arccos \left ( x \right ) \LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) +2\,\lambda \,\LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) \sqrt{-{x}^{2}+1}+2\,ax\sqrt{\arccos \left ( x \right ) }n+4\,ax\sqrt{\arccos \left ( x \right ) }}{ \left ( 2+n \right ) \sqrt{\arccos \left ( x \right ) }}}}}\,{\rm d}xa-\int \!{\frac{\LommelS 1 \left ( n+3/2,3/2,\arccos \left ( x \right ) \right ) }{\sqrt{\arccos \left ( x \right ) }}{{\rm e}^{-{\frac{-\LommelS 1 \left ( n+3/2,3/2,\arccos \left ( x \right ) \right ) \lambda \,\arccos \left ( x \right ) \sqrt{-{x}^{2}+1}-nx\lambda \,\arccos \left ( x \right ) \LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) +\lambda \,n\LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) \sqrt{-{x}^{2}+1}+\lambda \, \left ( \arccos \left ( x \right ) \right ) ^{n+3/2}\sqrt{-{x}^{2}+1}-2\,\lambda \,x\arccos \left ( x \right ) \LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) +2\,\lambda \,\LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) \sqrt{-{x}^{2}+1}+2\,ax\sqrt{\arccos \left ( x \right ) }n+4\,ax\sqrt{\arccos \left ( x \right ) }}{ \left ( 2+n \right ) \sqrt{\arccos \left ( x \right ) }}}}}}\,{\rm d}x\lambda +\int \!{\frac{\sqrt{\arccos \left ( x \right ) }\LommelS 1 \left ( n+3/2,3/2,\arccos \left ( x \right ) \right ) x}{\sqrt{-{x}^{2}+1}}{{\rm e}^{-{\frac{-\LommelS 1 \left ( n+3/2,3/2,\arccos \left ( x \right ) \right ) \lambda \,\arccos \left ( x \right ) \sqrt{-{x}^{2}+1}-nx\lambda \,\arccos \left ( x \right ) \LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) +\lambda \,n\LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) \sqrt{-{x}^{2}+1}+\lambda \, \left ( \arccos \left ( x \right ) \right ) ^{n+3/2}\sqrt{-{x}^{2}+1}-2\,\lambda \,x\arccos \left ( x \right ) \LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) +2\,\lambda \,\LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) \sqrt{-{x}^{2}+1}+2\,ax\sqrt{\arccos \left ( x \right ) }n+4\,ax\sqrt{\arccos \left ( x \right ) }}{ \left ( 2+n \right ) \sqrt{\arccos \left ( x \right ) }}}}}}\,{\rm d}x\lambda +\int \!{{\rm e}^{-{\frac{-\LommelS 1 \left ( n+3/2,3/2,\arccos \left ( x \right ) \right ) \lambda \,\arccos \left ( x \right ) \sqrt{-{x}^{2}+1}-nx\lambda \,\arccos \left ( x \right ) \LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) +\lambda \,n\LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) \sqrt{-{x}^{2}+1}+\lambda \, \left ( \arccos \left ( x \right ) \right ) ^{n+3/2}\sqrt{-{x}^{2}+1}-2\,\lambda \,x\arccos \left ( x \right ) \LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) +2\,\lambda \,\LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) \sqrt{-{x}^{2}+1}+2\,ax\sqrt{\arccos \left ( x \right ) }n+4\,ax\sqrt{\arccos \left ( x \right ) }}{ \left ( 2+n \right ) \sqrt{\arccos \left ( x \right ) }}}}} \left ( \arccos \left ( x \right ) \right ) ^{n}\,{\rm d}x\lambda -\int \!{\frac{ \left ( \arccos \left ( x \right ) \right ) ^{n+1}x}{\sqrt{-{x}^{2}+1}}{{\rm e}^{-{\frac{-\LommelS 1 \left ( n+3/2,3/2,\arccos \left ( x \right ) \right ) \lambda \,\arccos \left ( x \right ) \sqrt{-{x}^{2}+1}-nx\lambda \,\arccos \left ( x \right ) \LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) +\lambda \,n\LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) \sqrt{-{x}^{2}+1}+\lambda \, \left ( \arccos \left ( x \right ) \right ) ^{n+3/2}\sqrt{-{x}^{2}+1}-2\,\lambda \,x\arccos \left ( x \right ) \LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) +2\,\lambda \,\LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) \sqrt{-{x}^{2}+1}+2\,ax\sqrt{\arccos \left ( x \right ) }n+4\,ax\sqrt{\arccos \left ( x \right ) }}{ \left ( 2+n \right ) \sqrt{\arccos \left ( x \right ) }}}}}}\,{\rm d}x\lambda +n{{\rm e}^{{\frac{nx\lambda \,\arccos \left ( x \right ) \LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) +\LommelS 1 \left ( n+3/2,3/2,\arccos \left ( x \right ) \right ) \lambda \,\arccos \left ( x \right ) \sqrt{-{x}^{2}+1}-\lambda \,n\LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) \sqrt{-{x}^{2}+1}-2\,ax\sqrt{\arccos \left ( x \right ) }n+2\,\lambda \,x\arccos \left ( x \right ) \LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) -2\,\lambda \,\LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) \sqrt{-{x}^{2}+1}-\lambda \, \left ( \arccos \left ( x \right ) \right ) ^{n+3/2}\sqrt{-{x}^{2}+1}-4\,ax\sqrt{\arccos \left ( x \right ) }}{ \left ( 2+n \right ) \sqrt{\arccos \left ( x \right ) }}}}}+2\,\int \!{\frac{\LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) }{ \left ( \arccos \left ( x \right ) \right ) ^{3/2}}{{\rm e}^{-{\frac{-\LommelS 1 \left ( n+3/2,3/2,\arccos \left ( x \right ) \right ) \lambda \,\arccos \left ( x \right ) \sqrt{-{x}^{2}+1}-nx\lambda \,\arccos \left ( x \right ) \LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) +\lambda \,n\LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) \sqrt{-{x}^{2}+1}+\lambda \, \left ( \arccos \left ( x \right ) \right ) ^{n+3/2}\sqrt{-{x}^{2}+1}-2\,\lambda \,x\arccos \left ( x \right ) \LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) +2\,\lambda \,\LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) \sqrt{-{x}^{2}+1}+2\,ax\sqrt{\arccos \left ( x \right ) }n+4\,ax\sqrt{\arccos \left ( x \right ) }}{ \left ( 2+n \right ) \sqrt{\arccos \left ( x \right ) }}}}}}\,{\rm d}x\lambda -2\,\int \!{\frac{x\LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) }{\sqrt{\arccos \left ( x \right ) }\sqrt{-{x}^{2}+1}}{{\rm e}^{-{\frac{-\LommelS 1 \left ( n+3/2,3/2,\arccos \left ( x \right ) \right ) \lambda \,\arccos \left ( x \right ) \sqrt{-{x}^{2}+1}-nx\lambda \,\arccos \left ( x \right ) \LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) +\lambda \,n\LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) \sqrt{-{x}^{2}+1}+\lambda \, \left ( \arccos \left ( x \right ) \right ) ^{n+3/2}\sqrt{-{x}^{2}+1}-2\,\lambda \,x\arccos \left ( x \right ) \LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) +2\,\lambda \,\LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) \sqrt{-{x}^{2}+1}+2\,ax\sqrt{\arccos \left ( x \right ) }n+4\,ax\sqrt{\arccos \left ( x \right ) }}{ \left ( 2+n \right ) \sqrt{\arccos \left ( x \right ) }}}}}}\,{\rm d}x\lambda +2\,{{\rm e}^{{\frac{nx\lambda \,\arccos \left ( x \right ) \LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) +\LommelS 1 \left ( n+3/2,3/2,\arccos \left ( x \right ) \right ) \lambda \,\arccos \left ( x \right ) \sqrt{-{x}^{2}+1}-\lambda \,n\LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) \sqrt{-{x}^{2}+1}-2\,ax\sqrt{\arccos \left ( x \right ) }n+2\,\lambda \,x\arccos \left ( x \right ) \LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) -2\,\lambda \,\LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) \sqrt{-{x}^{2}+1}-\lambda \, \left ( \arccos \left ( x \right ) \right ) ^{n+3/2}\sqrt{-{x}^{2}+1}-4\,ax\sqrt{\arccos \left ( x \right ) }}{ \left ( 2+n \right ) \sqrt{\arccos \left ( x \right ) }}}}} \right ) } \right ) \]

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55.6 problem number 6

problem number 513

Added January 29, 2019.

Problem 2.7.2.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( y^2+\lambda x (\arccos x)^n y+ \lambda ( \arccos x)^n \right ) w_y = 0 \]

Mathematica

\[ \text{DSolve}\left [w^{(0,1)}(x,y) \left (a \lambda \cos ^{-1}(x)^n+\lambda x y \cos ^{-1}(x)^n+y^2\right )+w^{(1,0)}(x,y)=0,w(x,y),\{x,y\}\right ] \]

Maple

\[ \text{ sol=() } \]

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55.7 problem number 7

problem number 514

Added January 29, 2019.

Problem 2.7.2.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x - \left ( (k+1)x^k y^2 -\lambda (\arccos x)^n (x^{k+1} y-1) \right ) w_y = 0 \]

Mathematica

\[ \text{DSolve}\left [w^{(1,0)}(x,y)-w^{(0,1)}(x,y) \left ((k+1) y^2 x^k-\lambda \left (y x^{k+1}-1\right ) \cos ^{-1}(x)^n\right )=0,w(x,y),\{x,y\}\right ] \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac{1}{{x}^{k+1}y-1} \left ( -y{x}^{k+1}\int \!{\frac{{{\rm e}^{\lambda \,\int \! \left ( \arccos \left ( x \right ) \right ) ^{n}{x}^{k+1}\,{\rm d}x}}}{{x}^{k}{x}^{2}}}\,{\rm d}xk-y{x}^{k+1}\int \!{\frac{{{\rm e}^{\lambda \,\int \! \left ( \arccos \left ( x \right ) \right ) ^{n}{x}^{k+1}\,{\rm d}x}}}{{x}^{k}{x}^{2}}}\,{\rm d}x+{{\rm e}^{\int \!{\frac{ \left ( \arccos \left ( x \right ) \right ) ^{n}{x}^{k+1}\lambda \,x-2\,k-2}{x}}\,{\rm d}x}}{x}^{k+1}+\int \!{\frac{{{\rm e}^{\lambda \,\int \! \left ( \arccos \left ( x \right ) \right ) ^{n}{x}^{k+1}\,{\rm d}x}}}{{x}^{k}{x}^{2}}}\,{\rm d}xk+\int \!{\frac{{{\rm e}^{\lambda \,\int \! \left ( \arccos \left ( x \right ) \right ) ^{n}{x}^{k+1}\,{\rm d}x}}}{{x}^{k}{x}^{2}}}\,{\rm d}x \right ) } \right ) \]

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55.8 problem number 8

problem number 515

Added January 29, 2019.

Problem 2.7.2.8 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( \lambda (\arccos x)^n y^2+ a y+ a b - b^2 \lambda (\arccos x)^n \right ) w_y = 0 \]

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({\frac{1}{ \left ( b+y \right ) \left ( 2+n \right ) } \left ( 2\,\int \!{\frac{\sqrt{\arccos \left ( x \right ) }\LommelS 1 \left ( n-1/2,1/2,\arccos \left ( x \right ) \right ) x}{\sqrt{-{x}^{2}+1}}{{\rm e}^{{\frac{-2\,nbx\lambda \,\arccos \left ( x \right ) \LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) -2\,\LommelS 1 \left ( n+3/2,3/2,\arccos \left ( x \right ) \right ) b\lambda \,\arccos \left ( x \right ) \sqrt{-{x}^{2}+1}+2\,b\lambda \,n\LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) \sqrt{-{x}^{2}+1}-4\,b\lambda \,x\arccos \left ( x \right ) \LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) +4\,\lambda \,b\LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) \sqrt{-{x}^{2}+1}+2\,\lambda \,b \left ( \arccos \left ( x \right ) \right ) ^{n+3/2}\sqrt{-{x}^{2}+1}+ax\sqrt{\arccos \left ( x \right ) }n+2\,ax\sqrt{\arccos \left ( x \right ) }}{ \left ( 2+n \right ) \sqrt{\arccos \left ( x \right ) }}}}}}\,{\rm d}xb\lambda \,{n}^{2}-2\,\int \!{\frac{\LommelS 1 \left ( n-1/2,1/2,\arccos \left ( x \right ) \right ) }{\sqrt{\arccos \left ( x \right ) }}{{\rm e}^{{\frac{-2\,nbx\lambda \,\arccos \left ( x \right ) \LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) -2\,\LommelS 1 \left ( n+3/2,3/2,\arccos \left ( x \right ) \right ) b\lambda \,\arccos \left ( x \right ) \sqrt{-{x}^{2}+1}+2\,b\lambda \,n\LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) \sqrt{-{x}^{2}+1}-4\,b\lambda \,x\arccos \left ( x \right ) \LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) +4\,\lambda \,b\LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) \sqrt{-{x}^{2}+1}+2\,\lambda \,b \left ( \arccos \left ( x \right ) \right ) ^{n+3/2}\sqrt{-{x}^{2}+1}+ax\sqrt{\arccos \left ( x \right ) }n+2\,ax\sqrt{\arccos \left ( x \right ) }}{ \left ( 2+n \right ) \sqrt{\arccos \left ( x \right ) }}}}}}\,{\rm d}xb\lambda \,{n}^{2}+4\,\int \!{\frac{\sqrt{\arccos \left ( x \right ) }\LommelS 1 \left ( n-1/2,1/2,\arccos \left ( x \right ) \right ) x}{\sqrt{-{x}^{2}+1}}{{\rm e}^{{\frac{-2\,nbx\lambda \,\arccos \left ( x \right ) \LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) -2\,\LommelS 1 \left ( n+3/2,3/2,\arccos \left ( x \right ) \right ) b\lambda \,\arccos \left ( x \right ) \sqrt{-{x}^{2}+1}+2\,b\lambda \,n\LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) \sqrt{-{x}^{2}+1}-4\,b\lambda \,x\arccos \left ( x \right ) \LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) +4\,\lambda \,b\LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) \sqrt{-{x}^{2}+1}+2\,\lambda \,b \left ( \arccos \left ( x \right ) \right ) ^{n+3/2}\sqrt{-{x}^{2}+1}+ax\sqrt{\arccos \left ( x \right ) }n+2\,ax\sqrt{\arccos \left ( x \right ) }}{ \left ( 2+n \right ) \sqrt{\arccos \left ( x \right ) }}}}}}\,{\rm d}xb\lambda \,n-4\,\int \!{\frac{\LommelS 1 \left ( n-1/2,1/2,\arccos \left ( x \right ) \right ) }{\sqrt{\arccos \left ( x \right ) }}{{\rm e}^{{\frac{-2\,nbx\lambda \,\arccos \left ( x \right ) \LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) -2\,\LommelS 1 \left ( n+3/2,3/2,\arccos \left ( x \right ) \right ) b\lambda \,\arccos \left ( x \right ) \sqrt{-{x}^{2}+1}+2\,b\lambda \,n\LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) \sqrt{-{x}^{2}+1}-4\,b\lambda \,x\arccos \left ( x \right ) \LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) +4\,\lambda \,b\LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) \sqrt{-{x}^{2}+1}+2\,\lambda \,b \left ( \arccos \left ( x \right ) \right ) ^{n+3/2}\sqrt{-{x}^{2}+1}+ax\sqrt{\arccos \left ( x \right ) }n+2\,ax\sqrt{\arccos \left ( x \right ) }}{ \left ( 2+n \right ) \sqrt{\arccos \left ( x \right ) }}}}}}\,{\rm d}xb\lambda \,n-\int \!{{\rm e}^{{\frac{-2\,nbx\lambda \,\arccos \left ( x \right ) \LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) -2\,\LommelS 1 \left ( n+3/2,3/2,\arccos \left ( x \right ) \right ) b\lambda \,\arccos \left ( x \right ) \sqrt{-{x}^{2}+1}+2\,b\lambda \,n\LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) \sqrt{-{x}^{2}+1}-4\,b\lambda \,x\arccos \left ( x \right ) \LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) +4\,\lambda \,b\LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) \sqrt{-{x}^{2}+1}+2\,\lambda \,b \left ( \arccos \left ( x \right ) \right ) ^{n+3/2}\sqrt{-{x}^{2}+1}+ax\sqrt{\arccos \left ( x \right ) }n+2\,ax\sqrt{\arccos \left ( x \right ) }}{ \left ( 2+n \right ) \sqrt{\arccos \left ( x \right ) }}}}} \left ( \arccos \left ( x \right ) \right ) ^{n}\,{\rm d}xy\lambda \,n-\int \!{{\rm e}^{{\frac{-2\,nbx\lambda \,\arccos \left ( x \right ) \LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) -2\,\LommelS 1 \left ( n+3/2,3/2,\arccos \left ( x \right ) \right ) b\lambda \,\arccos \left ( x \right ) \sqrt{-{x}^{2}+1}+2\,b\lambda \,n\LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) \sqrt{-{x}^{2}+1}-4\,b\lambda \,x\arccos \left ( x \right ) \LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) +4\,\lambda \,b\LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) \sqrt{-{x}^{2}+1}+2\,\lambda \,b \left ( \arccos \left ( x \right ) \right ) ^{n+3/2}\sqrt{-{x}^{2}+1}+ax\sqrt{\arccos \left ( x \right ) }n+2\,ax\sqrt{\arccos \left ( x \right ) }}{ \left ( 2+n \right ) \sqrt{\arccos \left ( x \right ) }}}}} \left ( \arccos \left ( x \right ) \right ) ^{n}\,{\rm d}xb\lambda \,n+2\,\int \!{\frac{\LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) }{ \left ( \arccos \left ( x \right ) \right ) ^{3/2}}{{\rm e}^{{\frac{-2\,nbx\lambda \,\arccos \left ( x \right ) \LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) -2\,\LommelS 1 \left ( n+3/2,3/2,\arccos \left ( x \right ) \right ) b\lambda \,\arccos \left ( x \right ) \sqrt{-{x}^{2}+1}+2\,b\lambda \,n\LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) \sqrt{-{x}^{2}+1}-4\,b\lambda \,x\arccos \left ( x \right ) \LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) +4\,\lambda \,b\LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) \sqrt{-{x}^{2}+1}+2\,\lambda \,b \left ( \arccos \left ( x \right ) \right ) ^{n+3/2}\sqrt{-{x}^{2}+1}+ax\sqrt{\arccos \left ( x \right ) }n+2\,ax\sqrt{\arccos \left ( x \right ) }}{ \left ( 2+n \right ) \sqrt{\arccos \left ( x \right ) }}}}}}\,{\rm d}xb\lambda \,n-2\,\int \!{\frac{\LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) x}{\sqrt{\arccos \left ( x \right ) }\sqrt{-{x}^{2}+1}}{{\rm e}^{{\frac{-2\,nbx\lambda \,\arccos \left ( x \right ) \LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) -2\,\LommelS 1 \left ( n+3/2,3/2,\arccos \left ( x \right ) \right ) b\lambda \,\arccos \left ( x \right ) \sqrt{-{x}^{2}+1}+2\,b\lambda \,n\LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) \sqrt{-{x}^{2}+1}-4\,b\lambda \,x\arccos \left ( x \right ) \LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) +4\,\lambda \,b\LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) \sqrt{-{x}^{2}+1}+2\,\lambda \,b \left ( \arccos \left ( x \right ) \right ) ^{n+3/2}\sqrt{-{x}^{2}+1}+ax\sqrt{\arccos \left ( x \right ) }n+2\,ax\sqrt{\arccos \left ( x \right ) }}{ \left ( 2+n \right ) \sqrt{\arccos \left ( x \right ) }}}}}}\,{\rm d}xb\lambda \,n-2\,\int \!{{\rm e}^{{\frac{-2\,nbx\lambda \,\arccos \left ( x \right ) \LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) -2\,\LommelS 1 \left ( n+3/2,3/2,\arccos \left ( x \right ) \right ) b\lambda \,\arccos \left ( x \right ) \sqrt{-{x}^{2}+1}+2\,b\lambda \,n\LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) \sqrt{-{x}^{2}+1}-4\,b\lambda \,x\arccos \left ( x \right ) \LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) +4\,\lambda \,b\LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) \sqrt{-{x}^{2}+1}+2\,\lambda \,b \left ( \arccos \left ( x \right ) \right ) ^{n+3/2}\sqrt{-{x}^{2}+1}+ax\sqrt{\arccos \left ( x \right ) }n+2\,ax\sqrt{\arccos \left ( x \right ) }}{ \left ( 2+n \right ) \sqrt{\arccos \left ( x \right ) }}}}} \left ( \arccos \left ( x \right ) \right ) ^{n}\,{\rm d}xy\lambda +4\,\int \!{\frac{\LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) }{ \left ( \arccos \left ( x \right ) \right ) ^{3/2}}{{\rm e}^{{\frac{-2\,nbx\lambda \,\arccos \left ( x \right ) \LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) -2\,\LommelS 1 \left ( n+3/2,3/2,\arccos \left ( x \right ) \right ) b\lambda \,\arccos \left ( x \right ) \sqrt{-{x}^{2}+1}+2\,b\lambda \,n\LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) \sqrt{-{x}^{2}+1}-4\,b\lambda \,x\arccos \left ( x \right ) \LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) +4\,\lambda \,b\LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) \sqrt{-{x}^{2}+1}+2\,\lambda \,b \left ( \arccos \left ( x \right ) \right ) ^{n+3/2}\sqrt{-{x}^{2}+1}+ax\sqrt{\arccos \left ( x \right ) }n+2\,ax\sqrt{\arccos \left ( x \right ) }}{ \left ( 2+n \right ) \sqrt{\arccos \left ( x \right ) }}}}}}\,{\rm d}xb\lambda -2\,\int \!{\frac{ \left ( \arccos \left ( x \right ) \right ) ^{n+1}x}{\sqrt{-{x}^{2}+1}}{{\rm e}^{{\frac{-2\,nbx\lambda \,\arccos \left ( x \right ) \LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) -2\,\LommelS 1 \left ( n+3/2,3/2,\arccos \left ( x \right ) \right ) b\lambda \,\arccos \left ( x \right ) \sqrt{-{x}^{2}+1}+2\,b\lambda \,n\LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) \sqrt{-{x}^{2}+1}-4\,b\lambda \,x\arccos \left ( x \right ) \LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) +4\,\lambda \,b\LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) \sqrt{-{x}^{2}+1}+2\,\lambda \,b \left ( \arccos \left ( x \right ) \right ) ^{n+3/2}\sqrt{-{x}^{2}+1}+ax\sqrt{\arccos \left ( x \right ) }n+2\,ax\sqrt{\arccos \left ( x \right ) }}{ \left ( 2+n \right ) \sqrt{\arccos \left ( x \right ) }}}}}}\,{\rm d}xb\lambda -4\,\int \!{\frac{\LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) x}{\sqrt{\arccos \left ( x \right ) }\sqrt{-{x}^{2}+1}}{{\rm e}^{{\frac{-2\,nbx\lambda \,\arccos \left ( x \right ) \LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) -2\,\LommelS 1 \left ( n+3/2,3/2,\arccos \left ( x \right ) \right ) b\lambda \,\arccos \left ( x \right ) \sqrt{-{x}^{2}+1}+2\,b\lambda \,n\LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) \sqrt{-{x}^{2}+1}-4\,b\lambda \,x\arccos \left ( x \right ) \LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) +4\,\lambda \,b\LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) \sqrt{-{x}^{2}+1}+2\,\lambda \,b \left ( \arccos \left ( x \right ) \right ) ^{n+3/2}\sqrt{-{x}^{2}+1}+ax\sqrt{\arccos \left ( x \right ) }n+2\,ax\sqrt{\arccos \left ( x \right ) }}{ \left ( 2+n \right ) \sqrt{\arccos \left ( x \right ) }}}}}}\,{\rm d}xb\lambda -2\,\int \!{\frac{\LommelS 1 \left ( n+3/2,3/2,\arccos \left ( x \right ) \right ) }{\sqrt{\arccos \left ( x \right ) }}{{\rm e}^{{\frac{-2\,nbx\lambda \,\arccos \left ( x \right ) \LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) -2\,\LommelS 1 \left ( n+3/2,3/2,\arccos \left ( x \right ) \right ) b\lambda \,\arccos \left ( x \right ) \sqrt{-{x}^{2}+1}+2\,b\lambda \,n\LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) \sqrt{-{x}^{2}+1}-4\,b\lambda \,x\arccos \left ( x \right ) \LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) +4\,\lambda \,b\LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) \sqrt{-{x}^{2}+1}+2\,\lambda \,b \left ( \arccos \left ( x \right ) \right ) ^{n+3/2}\sqrt{-{x}^{2}+1}+ax\sqrt{\arccos \left ( x \right ) }n+2\,ax\sqrt{\arccos \left ( x \right ) }}{ \left ( 2+n \right ) \sqrt{\arccos \left ( x \right ) }}}}}}\,{\rm d}xb\lambda +2\,\int \!{\frac{\sqrt{\arccos \left ( x \right ) }\LommelS 1 \left ( n+3/2,3/2,\arccos \left ( x \right ) \right ) x}{\sqrt{-{x}^{2}+1}}{{\rm e}^{{\frac{-2\,nbx\lambda \,\arccos \left ( x \right ) \LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) -2\,\LommelS 1 \left ( n+3/2,3/2,\arccos \left ( x \right ) \right ) b\lambda \,\arccos \left ( x \right ) \sqrt{-{x}^{2}+1}+2\,b\lambda \,n\LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) \sqrt{-{x}^{2}+1}-4\,b\lambda \,x\arccos \left ( x \right ) \LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) +4\,\lambda \,b\LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) \sqrt{-{x}^{2}+1}+2\,\lambda \,b \left ( \arccos \left ( x \right ) \right ) ^{n+3/2}\sqrt{-{x}^{2}+1}+ax\sqrt{\arccos \left ( x \right ) }n+2\,ax\sqrt{\arccos \left ( x \right ) }}{ \left ( 2+n \right ) \sqrt{\arccos \left ( x \right ) }}}}}}\,{\rm d}xb\lambda -n{{\rm e}^{-{\frac{2\,\LommelS 1 \left ( n+3/2,3/2,\arccos \left ( x \right ) \right ) b\lambda \,\arccos \left ( x \right ) \sqrt{-{x}^{2}+1}+2\,nbx\lambda \,\arccos \left ( x \right ) \LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) -2\,\lambda \,b \left ( \arccos \left ( x \right ) \right ) ^{n+3/2}\sqrt{-{x}^{2}+1}-2\,b\lambda \,n\LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) \sqrt{-{x}^{2}+1}+4\,b\lambda \,x\arccos \left ( x \right ) \LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) -4\,\lambda \,b\LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) \sqrt{-{x}^{2}+1}-ax\sqrt{\arccos \left ( x \right ) }n-2\,ax\sqrt{\arccos \left ( x \right ) }}{ \left ( 2+n \right ) \sqrt{\arccos \left ( x \right ) }}}}}-2\,{{\rm e}^{-{\frac{2\,\LommelS 1 \left ( n+3/2,3/2,\arccos \left ( x \right ) \right ) b\lambda \,\arccos \left ( x \right ) \sqrt{-{x}^{2}+1}+2\,nbx\lambda \,\arccos \left ( x \right ) \LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) -2\,\lambda \,b \left ( \arccos \left ( x \right ) \right ) ^{n+3/2}\sqrt{-{x}^{2}+1}-2\,b\lambda \,n\LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) \sqrt{-{x}^{2}+1}+4\,b\lambda \,x\arccos \left ( x \right ) \LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) -4\,\lambda \,b\LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) \sqrt{-{x}^{2}+1}-ax\sqrt{\arccos \left ( x \right ) }n-2\,ax\sqrt{\arccos \left ( x \right ) }}{ \left ( 2+n \right ) \sqrt{\arccos \left ( x \right ) }}}}} \right ) } \right ) \]

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55.9 problem number 9

problem number 516

Added January 29, 2019.

Problem 2.7.2.9 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( \lambda (\arccos x)^n y^2- b \lambda x^m (\arccos x)^n y + b m x^{m-1} \right ) w_y = 0 \]

Mathematica

\[ \text{DSolve}\left [w^{(0,1)}(x,y) \left (-b \lambda y x^m \cos ^{-1}(x)^n+b m x^{m-1}+\lambda y^2 \cos ^{-1}(x)^n\right )+w^{(1,0)}(x,y)=0,w(x,y),\{x,y\}\right ] \]

Maple

\[ \text{ sol=() } \]

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55.10 problem number 10

problem number 517

Added January 29, 2019.

Problem 2.7.2.10 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( \lambda (\arccos x)^n y^2+ b m x^{m-1} - \lambda b^2 x^{2 m} (\arccos x)^n \right ) w_y = 0 \]

Mathematica

\[ \text{DSolve}\left [w^{(0,1)}(x,y) \left (-b^2 \lambda x^{2 m} \cos ^{-1}(x)^n+b m x^{m-1}+\lambda y^2 \cos ^{-1}(x)^n\right )+w^{(1,0)}(x,y)=0,w(x,y),\{x,y\}\right ] \]

Maple

\[ \text{ sol=() } \]

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55.11 problem number 11

problem number 518

Added January 29, 2019.

Problem 2.7.2.11 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( \lambda (\arccos x)^n (y- a x^m-b)^2 + a m x^{m-1} \right ) w_y = 0 \]

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({\frac{- \left ( \arccos \left ( x \right ) \right ) ^{n}\sqrt{-{x}^{2}+1} \left ( \arccos \left ( x \right ) \right ) ^{3/2}{x}^{m}a\lambda -\lambda \,b \left ( \arccos \left ( x \right ) \right ) ^{n} \left ( \arccos \left ( x \right ) \right ) ^{3/2}\sqrt{-{x}^{2}+1}+ \left ( \arccos \left ( x \right ) \right ) ^{n}\sqrt{-{x}^{2}+1} \left ( \arccos \left ( x \right ) \right ) ^{3/2}y\lambda +\LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ){x}^{m}\arccos \left ( x \right ) a\lambda \,nx+\sqrt{-{x}^{2}+1}\arccos \left ( x \right ) \LommelS 1 \left ( n+3/2,3/2,\arccos \left ( x \right ) \right ){x}^{m}a\lambda -\sqrt{-{x}^{2}+1}\LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ){x}^{m}a\lambda \,n+2\,\LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ){x}^{m}\arccos \left ( x \right ) a\lambda \,x+nbx\lambda \,\arccos \left ( x \right ) \LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) -\LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) y\arccos \left ( x \right ) \lambda \,nx+\LommelS 1 \left ( n+3/2,3/2,\arccos \left ( x \right ) \right ) b\lambda \,\arccos \left ( x \right ) \sqrt{-{x}^{2}+1}-\sqrt{-{x}^{2}+1}\LommelS 1 \left ( n+3/2,3/2,\arccos \left ( x \right ) \right ) y\arccos \left ( x \right ) \lambda -2\,\sqrt{-{x}^{2}+1}\LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ){x}^{m}a\lambda -b\lambda \,n\LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) \sqrt{-{x}^{2}+1}+\sqrt{-{x}^{2}+1}\LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) y\lambda \,n+2\,b\lambda \,x\arccos \left ( x \right ) \LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) -2\,\LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) y\arccos \left ( x \right ) \lambda \,x-2\,\lambda \,b\LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) \sqrt{-{x}^{2}+1}+2\,\sqrt{-{x}^{2}+1}\LommelS 1 \left ( 1/2+n,1/2,\arccos \left ( x \right ) \right ) y\lambda -\sqrt{\arccos \left ( x \right ) }n-2\,\sqrt{\arccos \left ( x \right ) }}{\sqrt{\arccos \left ( x \right ) } \left ({x}^{m}an+2\,a{x}^{m}-ny+nb-2\,y+2\,b \right ) }} \right ) \]

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55.12 problem number 12

problem number 519

Added January 29, 2019.

Problem 2.7.2.12 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ x w_x + \left ( \lambda (\arccos x)^n y^2+ k y + \lambda b^2 x^{2 k} (\arccos x)^n \right ) w_y = 0 \]

Mathematica

\[ \left \{\left \{w(x,y)\to c_1\left (\tan ^{-1}\left (\frac{y x^{-k}}{\sqrt{b^2}}\right )-\sqrt{b^2} \int _1^x \lambda K[1]^{k-1} \cos ^{-1}(K[1])^n \, dK[1]\right )\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( \lambda \,b\int \!{x}^{k-1} \left ( \arccos \left ( x \right ) \right ) ^{n}\,{\rm d}x-\arctan \left ({\frac{{x}^{-k}y}{b}} \right ) \right ) \]