143 HFOPDE, chapter 5.7.1

 143.1 Problem 1
 143.2 Problem 2
 143.3 Problem 3
 143.4 Problem 4
 143.5 Problem 5

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143.1 Problem 1

problem number 1132

Added April 13, 2019.

Problem Chapter 5.7.1.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b w_y = w + c_1 \arcsin ^k(\lambda x) + c_2 \arcsin ^n(\beta y) \]

Mathematica

\[ \text{\$Aborted} \]

Maple

\[ w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac{1}{a}{{\rm e}^{-{\frac{{\it \_a}}{a}}}} \left ({\it c2}\, \left ( \arcsin \left ({\frac{\beta \, \left ( ya-b \left ( x-{\it \_a} \right ) \right ) }{a}} \right ) \right ) ^{n}+{\it c1}\, \left ( \arcsin \left ( \lambda \,{\it \_a} \right ) \right ) ^{k} \right ) }{d{\it \_a}}+{\it \_F1} \left ({\frac{ya-bx}{a}} \right ) \right ){{\rm e}^{{\frac{x}{a}}}} \]

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143.2 Problem 2

problem number 1133

Added April 13, 2019.

Problem Chapter 5.7.1.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b w_y = c w + \arcsin ^k(\lambda x) \arcsin ^n(\beta y) \]

Mathematica

\[ \text{\$Aborted} \]

Maple

\[ w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac{ \left ( \arcsin \left ( \lambda \,{\it \_a} \right ) \right ) ^{k}}{a} \left ( \arcsin \left ({\frac{\beta \, \left ( ya-b \left ( x-{\it \_a} \right ) \right ) }{a}} \right ) \right ) ^{n}{{\rm e}^{-{\frac{{\it \_a}\,c}{a}}}}}{d{\it \_a}}+{\it \_F1} \left ({\frac{ya-bx}{a}} \right ) \right ){{\rm e}^{{\frac{cx}{a}}}} \]

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143.3 Problem 3

problem number 1134

Added April 13, 2019.

Problem Chapter 5.7.1.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b w_y = \left ( c_1 \arcsin (\lambda _1 x) + c_2 \arcsin (\lambda _2 y)\right ) w+ s_1 \arcsin ^n(\beta _1 x)+ s_2 \arcsin ^k(\beta _2 y) \]

Mathematica

\[ \text{\$Aborted} \]

Maple

\[ w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac{1}{a}{{\rm e}^{{\frac{1}{a\lambda 1\,\lambda 2\,b} \left ( -\sqrt{-{\frac{ \left ( \left ( \lambda 2\,y-1 \right ) a-b\lambda 2\, \left ( x-{\it \_a} \right ) \right ) \left ( \left ( \lambda 2\,y+1 \right ) a-b\lambda 2\, \left ( x-{\it \_a} \right ) \right ) }{{a}^{2}}}}a{\it c2}\,\lambda 1- \left ( \left ( \left ({\it \_a}-x \right ) b+ya \right ){\it c2}\,\lambda 1\,\arcsin \left ({\frac{\lambda 2\, \left ( ya-b \left ( x-{\it \_a} \right ) \right ) }{a}} \right ) +b{\it c1}\, \left ( \arcsin \left ( \lambda 1\,{\it \_a} \right ){\it \_a}\,\lambda 1+\sqrt{-{{\it \_a}}^{2}{\lambda 1}^{2}+1} \right ) \right ) \lambda 2 \right ) }}} \left ({\it s1}\, \left ( \arcsin \left ( \beta 1\,{\it \_a} \right ) \right ) ^{n}+{\it s2}\, \left ( \arcsin \left ({\frac{\beta 2\, \left ( ya-b \left ( x-{\it \_a} \right ) \right ) }{a}} \right ) \right ) ^{k} \right ) }{d{\it \_a}}+{\it \_F1} \left ({\frac{ya-bx}{a}} \right ) \right ){{\rm e}^{{\frac{\sqrt{-{\lambda 2}^{2}{y}^{2}+1}a{\it c2}\,\lambda 1+ \left ( b{\it c1}\,\sqrt{-{\lambda 1}^{2}{x}^{2}+1}+\lambda 1\, \left ( a\arcsin \left ( \lambda 2\,y \right ) y{\it c2}+bx{\it c1}\,\arcsin \left ( \lambda 1\,x \right ) \right ) \right ) \lambda 2}{a\lambda 1\,\lambda 2\,b}}}} \]

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143.4 Problem 4

problem number 1135

Added April 13, 2019.

Problem Chapter 5.7.1.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b \arcsin ^m(\mu x) w_y = c \arcsin ^k(\nu x) w + p \arcsin ^n(\beta y) \]

Mathematica

\[ \text{\$Aborted} \]

Maple

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

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143.5 Problem 5

problem number 1136

Added April 13, 2019.

Problem Chapter 5.7.1.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b \arcsin ^m(\mu x) w_y = c \arcsin ^k(\nu y) w + p \arcsin ^n(\beta x) \]

Mathematica

\[ \text{\$Aborted} \]

Maple

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