7.2.20 7.1

7.2.20.1 [705] problem number 1

problem number 705

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 ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (a*ArcSin[lambda*x]^k + b)*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

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

Maple ✓

restart; 
pde :=  diff(w(x,y),x)+ (a*arcsin(lambda*x)^k+b)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

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

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7.2.20.2 [706] problem number 2

problem number 706

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 ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (a*ArcSin[lambda*y]^k + b)*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

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

Maple ✓

restart; 
pde :=  diff(w(x,y),x)+ (a*arcsin(lambda*y)^k+b)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

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

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7.2.20.3 [707] problem number 3

problem number 707

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 ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + k*Arcsin[a*x + b*y + c]^n*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple ✓

restart; 
pde :=  diff(w(x,y),x)+ k*arcsin(a*x + b*y+c)^n*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

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

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7.2.20.4 [708] problem number 4

problem number 708

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 ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + a*Arcsin[lambda*x]^k*Arcsin[mu*y]^n*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

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

Maple ✓

restart; 
pde :=  diff(w(x,y),x)+a*arcsin(lambda*x)^k*arcsin(mu*y)^n*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

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

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7.2.20.5 [709] problem number 5

problem number 709

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 ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (y^2 + lambda*Arcsin[x]^n*y - a^2 + a*lambda*Arcsin[x]^n)*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple ✓

restart; 
pde :=  diff(w(x,y),x)+(y^2+ lambda*arcsin(x)^n*y -a^2 + a *lambda*arcsin(x)^n)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = \textit {\_F1} \left (\frac {\left (-a -y \right ) \left (\int {\mathrm e}^{-\frac {2 \left (x +1\right ) \left (x -1\right ) \left (\frac {\left (-\arcsin \left (x \right )^{n}+\frac {\LommelS 1 \left (n +\frac {3}{2}, \frac {1}{2}, \arcsin \left (x \right )\right )}{\sqrt {\arcsin \left (x \right )}}\right ) \sqrt {-x^{2}+1}\, \lambda 2^{n} 2^{-n} \arcsin \left (x \right )}{2}+\left (-\frac {\lambda n 2^{n} 2^{-n} \LommelS 1 \left (n +\frac {1}{2}, \frac {3}{2}, \arcsin \left (x \right )\right ) \sqrt {\arcsin \left (x \right )}}{2}+\frac {\lambda 2^{n} 2^{-n} \arcsin \left (x \right )^{n}}{2}-\lambda 2^{n} 2^{-n -1} \arcsin \left (x \right )^{n}-\frac {\lambda 2^{n} 2^{-n} \LommelS 1 \left (n +\frac {3}{2}, \frac {1}{2}, \arcsin \left (x \right )\right )}{2 \sqrt {\arcsin \left (x \right )}}+\left (n +1\right ) a \right ) x \right )}{\left (n +1\right ) x^{2}-n -1}}d x \right )-{\mathrm e}^{-\frac {2 \left (x +1\right ) \left (x -1\right ) \left (\frac {\left (-\arcsin \left (x \right )^{n}+\frac {\LommelS 1 \left (n +\frac {3}{2}, \frac {1}{2}, \arcsin \left (x \right )\right )}{\sqrt {\arcsin \left (x \right )}}\right ) \sqrt {-x^{2}+1}\, \lambda 2^{n} 2^{-n} \arcsin \left (x \right )}{2}+\left (-\frac {\lambda n 2^{n} 2^{-n} \LommelS 1 \left (n +\frac {1}{2}, \frac {3}{2}, \arcsin \left (x \right )\right ) \sqrt {\arcsin \left (x \right )}}{2}+\frac {\lambda 2^{n} 2^{-n} \arcsin \left (x \right )^{n}}{2}-\lambda 2^{n} 2^{-n -1} \arcsin \left (x \right )^{n}-\frac {\lambda 2^{n} 2^{-n} \LommelS 1 \left (n +\frac {3}{2}, \frac {1}{2}, \arcsin \left (x \right )\right )}{2 \sqrt {\arcsin \left (x \right )}}+\left (n +1\right ) a \right ) x \right )}{\left (n +1\right ) x^{2}-n -1}}}{a +y}\right )\]

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7.2.20.6 [710] problem number 6

problem number 710

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 ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (y^2 + lambda*x*Arcsin[x]^n*y + lambda*Arcsin[x]^n)*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to c_1\left (-\frac {\exp \left (-\int _1^x-\lambda \text {Arcsin}(K[5])^n K[5]dK[5]\right )}{x^2 y+x}-\int _1^x\frac {\exp \left (-\int _1^{K[6]}-\lambda \text {Arcsin}(K[5])^n K[5]dK[5]\right )}{K[6]^2}dK[6]\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=  diff(w(x,y),x)+( y^2+ lambda*x*arcsin(x)^n*y + lambda*arcsin(x)^n)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

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

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7.2.20.7 [711] problem number 7

problem number 711

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 ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] - ((k + 1)*x^k*y^2 - lambda*Arcsin[x]^n*(x^(k + 1)*y - 1))*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple ✓

restart; 
pde :=  diff(w(x,y),x)-(  (k+1)*x^k*y^2 - lambda*arcsin(x)^n*(x^(k+1)*y-1))*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

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

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7.2.20.8 [712] problem number 8

problem number 712

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 ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (lambda*Arcsin[x]^n*y^2 + a*y + a*b - b^2*lambda*Arcsin[x]^n)*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple ✓

restart; 
pde :=  diff(w(x,y),x)+( lambda*arcsin(x)^n*y^2 + a*y+ a*b -b^2 * lambda*arcsin(x)^n)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = \textit {\_F1} \left (\frac {-\left (b +y \right ) \lambda \left (\int \arcsin \left (x \right )^{n} {\mathrm e}^{\frac {\left (x +1\right ) \left (-2 \left (\arcsin \left (x \right )^{n}-\frac {\LommelS 1 \left (n +\frac {3}{2}, \frac {1}{2}, \arcsin \left (x \right )\right )}{\sqrt {\arcsin \left (x \right )}}\right ) \sqrt {-x^{2}+1}\, b \lambda 2^{n} 2^{-n} \arcsin \left (x \right )+\left (-2 b \lambda n 2^{n} 2^{-n} \LommelS 1 \left (n +\frac {1}{2}, \frac {3}{2}, \arcsin \left (x \right )\right ) \sqrt {\arcsin \left (x \right )}+2 b \lambda 2^{n} 2^{-n} \arcsin \left (x \right )^{n}-4 b \lambda 2^{n} 2^{-n -1} \arcsin \left (x \right )^{n}-\frac {2 b \lambda 2^{n} 2^{-n} \LommelS 1 \left (n +\frac {3}{2}, \frac {1}{2}, \arcsin \left (x \right )\right )}{\sqrt {\arcsin \left (x \right )}}+\left (n +1\right ) a \right ) x \right ) \left (x -1\right )}{\left (n +1\right ) x^{2}-n -1}}d x \right )-{\mathrm e}^{\frac {\left (x +1\right ) \left (-2 \left (\arcsin \left (x \right )^{n}-\frac {\LommelS 1 \left (n +\frac {3}{2}, \frac {1}{2}, \arcsin \left (x \right )\right )}{\sqrt {\arcsin \left (x \right )}}\right ) \sqrt {-x^{2}+1}\, b \lambda 2^{n} 2^{-n} \arcsin \left (x \right )+\left (-2 b \lambda n 2^{n} 2^{-n} \LommelS 1 \left (n +\frac {1}{2}, \frac {3}{2}, \arcsin \left (x \right )\right ) \sqrt {\arcsin \left (x \right )}+2 b \lambda 2^{n} 2^{-n} \arcsin \left (x \right )^{n}-4 b \lambda 2^{n} 2^{-n -1} \arcsin \left (x \right )^{n}-\frac {2 b \lambda 2^{n} 2^{-n} \LommelS 1 \left (n +\frac {3}{2}, \frac {1}{2}, \arcsin \left (x \right )\right )}{\sqrt {\arcsin \left (x \right )}}+\left (n +1\right ) a \right ) x \right ) \left (x -1\right )}{\left (n +1\right ) x^{2}-n -1}}}{b +y}\right )\]

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7.2.20.9 [713] problem number 9

problem number 713

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 ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (lambda*Arcsin[x]^n*y^2 - b*lambda*x^m*ArcSin[x]^n*y + b*m*x^(m - 1))*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple ✗

restart; 
pde :=  diff(w(x,y),x)+( lambda*arcsin(x)^n*y^2 - b*lambda*x^m*arcsin(x)^n*y+b*m*x^(m-1) )*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

sol=()

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7.2.20.10 [714] problem number 10

problem number 714

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 ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (lambda*ArcSin[x]^n*y^2 + b*m*x^(m - 1) - lambda*b^2*x^(2*m)*ArcSin[x]^n)*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple ✗

restart; 
pde :=  diff(w(x,y),x)+( lambda*arcsin(x)^n*y^2 + b*m*x^(m-1) - lambda*b^2*x^(2*m)*arcsin(x)^n  )*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

sol=()

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7.2.20.11 [715] problem number 11

problem number 715

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 ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (lambda*ArcSin[x]^n*(y - a*x^m - b)^2 + a*m*x^(m - 1))*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

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

Maple ✓

restart; 
pde :=  diff(w(x,y),x)+( lambda*arcsin(x)^n*(y - a*x^m -b)^2 + a*m*x^(m-1)  )*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

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

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7.2.20.12 [716] problem number 12

problem number 716

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 ✓

ClearAll["Global`*"]; 
pde =  x*D[w[x, y], x] + (lambda*ArcSin[x]^n*y^2 + k*y + lambda*b^2*x^(2*k)*ArcSin[x]^n)*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

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

Maple ✓

restart; 
pde :=  x*diff(w(x,y),x)+( lambda*arcsin(x)^n*y^2 +k*y+ lambda*b^2*x^(2*k)*arcsin(x)^n  )*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

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

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