54 HFOPDE, chapter 2.7.1

 54.1 problem number 1
 54.2 problem number 2
 54.3 problem number 3
 54.4 problem number 4
 54.5 problem number 5
 54.6 problem number 6
 54.7 problem number 7
 54.8 problem number 8
 54.9 problem number 9
 54.10 problem number 10
 54.11 problem number 11
 54.12 problem number 12

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54.1 problem number 1

problem number 496

Added January 20, 2019.

Problem 2.7.1.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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54.2 problem number 2

problem number 497

Added January 20, 2019.

Problem 2.7.1.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

\[ \text{Timed out} \]

Maple

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

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54.3 problem number 3

problem number 498

Added January 20, 2019.

Problem 2.7.1.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

\[ \text{DSolve}\left [k w^{(0,1)}(x,y) \text{Arcsin}(a x+b y+c)^n+w^{(1,0)}(x,y)=0,w(x,y),\{x,y\}\right ] \]

Maple

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

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54.4 problem number 4

problem number 499

Added January 20, 2019.

Problem 2.7.1.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

\[ \left \{\left \{w(x,y)\to c_1\left (\int _1^y \text{Arcsin}(\mu K[1])^{-n} \, dK[1]-\int _1^x a \text{Arcsin}(\lambda K[2])^k \, dK[2]\right )\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac{{2}^{k}\sqrt{\pi }}{\lambda } \left ({\frac{{2}^{-1-k} \left ( \arcsin \left ( \lambda \,x \right ) \right ) ^{k} \left ( 6+2\,k \right ) \lambda \,x}{\sqrt{\pi } \left ( k+1 \right ) \left ( 3+k \right ) }}+{\frac{ \left ( \arcsin \left ( \lambda \,x \right ) \right ) ^{k}{2}^{-k}\sqrt{-{\lambda }^{2}{x}^{2}+1} \left ( \arcsin \left ( \lambda \,x \right ){x}^{2}{\lambda }^{2}-\arcsin \left ( \lambda \,x \right ) +\sqrt{-{\lambda }^{2}{x}^{2}+1}x\lambda \right ) }{\sqrt{\pi } \left ( k+1 \right ) \left ({\lambda }^{2}{x}^{2}-1 \right ) }}+{\frac{{2}^{-k}\sqrt{\arcsin \left ( \lambda \,x \right ) }k\LommelS 1 \left ( k+1/2,3/2,\arcsin \left ( \lambda \,x \right ) \right ) \lambda \,x}{\sqrt{\pi } \left ( k+1 \right ) }}-{\frac{{2}^{-k}\sqrt{-{\lambda }^{2}{x}^{2}+1} \left ( \arcsin \left ( \lambda \,x \right ){x}^{2}{\lambda }^{2}-\arcsin \left ( \lambda \,x \right ) +\sqrt{-{\lambda }^{2}{x}^{2}+1}x\lambda \right ) \LommelS 1 \left ( k+3/2,1/2,\arcsin \left ( \lambda \,x \right ) \right ) }{\sqrt{\pi } \left ( k+1 \right ) \sqrt{\arcsin \left ( \lambda \,x \right ) } \left ({\lambda }^{2}{x}^{2}-1 \right ) }} \right ) }-{\frac{{2}^{-n} \left ( \left ( \arcsin \left ( \mu \,y \right ) \right ) ^{-n}\sqrt{-{\mu }^{2}{y}^{2}+1} \left ( \arcsin \left ( \mu \,y \right ) \right ) ^{3/2}{2}^{n}-{2}^{n}\arcsin \left ( \mu \,y \right ) n\LommelS 1 \left ( -n+1/2,3/2,\arcsin \left ( \mu \,y \right ) \right ) \mu \,y- \left ( \arcsin \left ( \mu \,y \right ) \right ) ^{-n}\sqrt{\arcsin \left ( \mu \,y \right ) }{2}^{n}y\mu +2\,{2}^{n-1} \left ( \arcsin \left ( \mu \,y \right ) \right ) ^{-n}\mu \,y\sqrt{\arcsin \left ( \mu \,y \right ) }-\sqrt{-{\mu }^{2}{y}^{2}+1}\arcsin \left ( \mu \,y \right ) \LommelS 1 \left ( -n+3/2,1/2,\arcsin \left ( \mu \,y \right ) \right ){2}^{n}+\LommelS 1 \left ( -n+3/2,1/2,\arcsin \left ( \mu \,y \right ) \right ){2}^{n}y\mu \right ) }{ \left ( n-1 \right ) \mu \,a\sqrt{\arcsin \left ( \mu \,y \right ) }}} \right ) \]

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54.5 problem number 5

problem number 500

Added January 20, 2019.

Problem 2.7.1.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac{1}{a+y} \left ( y\int \!{{\rm e}^{\lambda \,{2}^{n}\sqrt{\pi } \left ({\frac{{2}^{-n-1} \left ( \arcsin \left ( x \right ) \right ) ^{n} \left ( 6+2\,n \right ) x}{\sqrt{\pi } \left ( n+1 \right ) \left ( n+3 \right ) }}+{\frac{ \left ( \arcsin \left ( x \right ) \right ) ^{n}{2}^{-n}\sqrt{-{x}^{2}+1} \left ( \arcsin \left ( x \right ){x}^{2}-\arcsin \left ( x \right ) +\sqrt{-{x}^{2}+1}x \right ) }{\sqrt{\pi } \left ( n+1 \right ) \left ({x}^{2}-1 \right ) }}+{\frac{{2}^{-n}\sqrt{\arcsin \left ( x \right ) }n\LommelS 1 \left ( 1/2+n,3/2,\arcsin \left ( x \right ) \right ) x}{\sqrt{\pi } \left ( n+1 \right ) }}-{\frac{{2}^{-n}\sqrt{-{x}^{2}+1} \left ( \arcsin \left ( x \right ){x}^{2}-\arcsin \left ( x \right ) +\sqrt{-{x}^{2}+1}x \right ) \LommelS 1 \left ( n+3/2,1/2,\arcsin \left ( x \right ) \right ) }{\sqrt{\pi } \left ( n+1 \right ) \sqrt{\arcsin \left ( x \right ) } \left ({x}^{2}-1 \right ) }} \right ) -2\,ax}}\,{\rm d}x+\int \!{{\rm e}^{\lambda \,{2}^{n}\sqrt{\pi } \left ({\frac{{2}^{-n-1} \left ( \arcsin \left ( x \right ) \right ) ^{n} \left ( 6+2\,n \right ) x}{\sqrt{\pi } \left ( n+1 \right ) \left ( n+3 \right ) }}+{\frac{ \left ( \arcsin \left ( x \right ) \right ) ^{n}{2}^{-n}\sqrt{-{x}^{2}+1} \left ( \arcsin \left ( x \right ){x}^{2}-\arcsin \left ( x \right ) +\sqrt{-{x}^{2}+1}x \right ) }{\sqrt{\pi } \left ( n+1 \right ) \left ({x}^{2}-1 \right ) }}+{\frac{{2}^{-n}\sqrt{\arcsin \left ( x \right ) }n\LommelS 1 \left ( 1/2+n,3/2,\arcsin \left ( x \right ) \right ) x}{\sqrt{\pi } \left ( n+1 \right ) }}-{\frac{{2}^{-n}\sqrt{-{x}^{2}+1} \left ( \arcsin \left ( x \right ){x}^{2}-\arcsin \left ( x \right ) +\sqrt{-{x}^{2}+1}x \right ) \LommelS 1 \left ( n+3/2,1/2,\arcsin \left ( x \right ) \right ) }{\sqrt{\pi } \left ( n+1 \right ) \sqrt{\arcsin \left ( x \right ) } \left ({x}^{2}-1 \right ) }} \right ) -2\,ax}}\,{\rm d}xa+{{\rm e}^{\lambda \,{2}^{n}\sqrt{\pi } \left ({\frac{{2}^{-n-1} \left ( \arcsin \left ( x \right ) \right ) ^{n} \left ( 6+2\,n \right ) x}{\sqrt{\pi } \left ( n+1 \right ) \left ( n+3 \right ) }}+{\frac{ \left ( \arcsin \left ( x \right ) \right ) ^{n}{2}^{-n}\sqrt{-{x}^{2}+1} \left ( \arcsin \left ( x \right ){x}^{2}-\arcsin \left ( x \right ) +\sqrt{-{x}^{2}+1}x \right ) }{\sqrt{\pi } \left ( n+1 \right ) \left ({x}^{2}-1 \right ) }}+{\frac{{2}^{-n}\sqrt{\arcsin \left ( x \right ) }n\LommelS 1 \left ( 1/2+n,3/2,\arcsin \left ( x \right ) \right ) x}{\sqrt{\pi } \left ( n+1 \right ) }}-{\frac{{2}^{-n}\sqrt{-{x}^{2}+1} \left ( \arcsin \left ( x \right ){x}^{2}-\arcsin \left ( x \right ) +\sqrt{-{x}^{2}+1}x \right ) \LommelS 1 \left ( n+3/2,1/2,\arcsin \left ( x \right ) \right ) }{\sqrt{\pi } \left ( n+1 \right ) \sqrt{\arcsin \left ( x \right ) } \left ({x}^{2}-1 \right ) }} \right ) -2\,ax}} \right ) } \right ) \]

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54.6 problem number 6

problem number 501

Added January 20, 2019.

Problem 2.7.1.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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54.7 problem number 7

problem number 502

Added January 20, 2019.

Problem 2.7.1.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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54.8 problem number 8

problem number 503

Added January 20, 2019.

Problem 2.7.1.8 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac{1}{b+y} \left ( y\lambda \,\int \! \left ( \arcsin \left ( x \right ) \right ) ^{n}{{\rm e}^{-2\,\lambda \,b{2}^{n}\sqrt{\pi } \left ({\frac{{2}^{-n-1} \left ( \arcsin \left ( x \right ) \right ) ^{n} \left ( 6+2\,n \right ) x}{\sqrt{\pi } \left ( n+1 \right ) \left ( n+3 \right ) }}+{\frac{ \left ( \arcsin \left ( x \right ) \right ) ^{n}{2}^{-n}\sqrt{-{x}^{2}+1} \left ( \arcsin \left ( x \right ){x}^{2}-\arcsin \left ( x \right ) +\sqrt{-{x}^{2}+1}x \right ) }{\sqrt{\pi } \left ( n+1 \right ) \left ({x}^{2}-1 \right ) }}+{\frac{{2}^{-n}\sqrt{\arcsin \left ( x \right ) }n\LommelS 1 \left ( 1/2+n,3/2,\arcsin \left ( x \right ) \right ) x}{\sqrt{\pi } \left ( n+1 \right ) }}-{\frac{{2}^{-n}\sqrt{-{x}^{2}+1} \left ( \arcsin \left ( x \right ){x}^{2}-\arcsin \left ( x \right ) +\sqrt{-{x}^{2}+1}x \right ) \LommelS 1 \left ( n+3/2,1/2,\arcsin \left ( x \right ) \right ) }{\sqrt{\pi } \left ( n+1 \right ) \sqrt{\arcsin \left ( x \right ) } \left ({x}^{2}-1 \right ) }} \right ) +ax}}\,{\rm d}x+\lambda \,\int \! \left ( \arcsin \left ( x \right ) \right ) ^{n}{{\rm e}^{-2\,\lambda \,b{2}^{n}\sqrt{\pi } \left ({\frac{{2}^{-n-1} \left ( \arcsin \left ( x \right ) \right ) ^{n} \left ( 6+2\,n \right ) x}{\sqrt{\pi } \left ( n+1 \right ) \left ( n+3 \right ) }}+{\frac{ \left ( \arcsin \left ( x \right ) \right ) ^{n}{2}^{-n}\sqrt{-{x}^{2}+1} \left ( \arcsin \left ( x \right ){x}^{2}-\arcsin \left ( x \right ) +\sqrt{-{x}^{2}+1}x \right ) }{\sqrt{\pi } \left ( n+1 \right ) \left ({x}^{2}-1 \right ) }}+{\frac{{2}^{-n}\sqrt{\arcsin \left ( x \right ) }n\LommelS 1 \left ( 1/2+n,3/2,\arcsin \left ( x \right ) \right ) x}{\sqrt{\pi } \left ( n+1 \right ) }}-{\frac{{2}^{-n}\sqrt{-{x}^{2}+1} \left ( \arcsin \left ( x \right ){x}^{2}-\arcsin \left ( x \right ) +\sqrt{-{x}^{2}+1}x \right ) \LommelS 1 \left ( n+3/2,1/2,\arcsin \left ( x \right ) \right ) }{\sqrt{\pi } \left ( n+1 \right ) \sqrt{\arcsin \left ( x \right ) } \left ({x}^{2}-1 \right ) }} \right ) +ax}}\,{\rm d}xb+{{\rm e}^{-2\,\lambda \,b{2}^{n}\sqrt{\pi } \left ({\frac{{2}^{-n-1} \left ( \arcsin \left ( x \right ) \right ) ^{n} \left ( 6+2\,n \right ) x}{\sqrt{\pi } \left ( n+1 \right ) \left ( n+3 \right ) }}+{\frac{ \left ( \arcsin \left ( x \right ) \right ) ^{n}{2}^{-n}\sqrt{-{x}^{2}+1} \left ( \arcsin \left ( x \right ){x}^{2}-\arcsin \left ( x \right ) +\sqrt{-{x}^{2}+1}x \right ) }{\sqrt{\pi } \left ( n+1 \right ) \left ({x}^{2}-1 \right ) }}+{\frac{{2}^{-n}\sqrt{\arcsin \left ( x \right ) }n\LommelS 1 \left ( 1/2+n,3/2,\arcsin \left ( x \right ) \right ) x}{\sqrt{\pi } \left ( n+1 \right ) }}-{\frac{{2}^{-n}\sqrt{-{x}^{2}+1} \left ( \arcsin \left ( x \right ){x}^{2}-\arcsin \left ( x \right ) +\sqrt{-{x}^{2}+1}x \right ) \LommelS 1 \left ( n+3/2,1/2,\arcsin \left ( x \right ) \right ) }{\sqrt{\pi } \left ( n+1 \right ) \sqrt{\arcsin \left ( x \right ) } \left ({x}^{2}-1 \right ) }} \right ) +ax}} \right ) } \right ) \]

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54.9 problem number 9

problem number 504

Added January 29, 2019.

Problem 2.7.1.9 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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54.10 problem number 10

problem number 505

Added January 29, 2019.

Problem 2.7.1.10 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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54.11 problem number 11

problem number 506

Added January 29, 2019.

Problem 2.7.1.11 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

\[ \text{Timed out} \]

Maple

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

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54.12 problem number 12

problem number 507

Added January 29, 2019.

Problem 2.7.1.12 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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