Internal problem ID [9818]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second
edition
Section: Chapter 1, section 1.2. Riccati Equation. subsection 1.2.7-1. Equations containing
arcsine.
Problem number: 4.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
Solve \begin {gather*} \boxed {y^{\prime }-\lambda \arcsin \relax (x )^{n} y^{2}-a y-a b +b^{2} \lambda \arcsin \relax (x )^{n}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.003 (sec). Leaf size: 114
dsolve(diff(y(x),x)=lambda*arcsin(x)^n*y(x)^2+a*y(x)+a*b-b^2*lambda*arcsin(x)^n,y(x), singsol=all)
\[ y \relax (x ) = -\frac {\left (\left (\int \arcsin \relax (x )^{n} \lambda \,{\mathrm e}^{\int \left (-2 \arcsin \relax (x )^{n} \lambda b +a \right )d x}d x \right ) {\mathrm e}^{\int \left (2 \arcsin \relax (x )^{n} \lambda b -a \right )d x} b +c_{1} {\mathrm e}^{\int \left (2 \arcsin \relax (x )^{n} \lambda b -a \right )d x} b +1\right ) {\mathrm e}^{\int \left (-2 \arcsin \relax (x )^{n} \lambda b +a \right )d x}}{c_{1}+\int \arcsin \relax (x )^{n} \lambda \,{\mathrm e}^{\int \left (-2 \arcsin \relax (x )^{n} \lambda b +a \right )d x}d x} \]
✓ Solution by Mathematica
Time used: 3.534 (sec). Leaf size: 428
DSolve[y'[x]==\[Lambda]*ArcSin[x]^n*y[x]^2+a*y[x]+a*b-b^2*\[Lambda]*ArcSin[x]^n,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^x\frac {i \exp \left (a K[1]-i b \lambda \text {ArcSin}(K[1])^n \left (\text {ArcSin}(K[1])^2\right )^{-n} \left ((-i \text {ArcSin}(K[1]))^n \text {Gamma}(n+1,i \text {ArcSin}(K[1]))-(i \text {ArcSin}(K[1]))^n \text {Gamma}(n+1,-i \text {ArcSin}(K[1]))\right )\right ) \left (-b \lambda \text {ArcSin}(K[1])^n+\lambda y(x) \text {ArcSin}(K[1])^n+a\right )}{a n \lambda (b+y(x))}dK[1]+\int _1^{y(x)}\left (-\int _1^x\left (\frac {i \exp \left (a K[1]-i b \lambda \text {ArcSin}(K[1])^n \left (\text {ArcSin}(K[1])^2\right )^{-n} \left ((-i \text {ArcSin}(K[1]))^n \text {Gamma}(n+1,i \text {ArcSin}(K[1]))-(i \text {ArcSin}(K[1]))^n \text {Gamma}(n+1,-i \text {ArcSin}(K[1]))\right )\right ) \text {ArcSin}(K[1])^n}{a n (b+K[2])}-\frac {i \exp \left (a K[1]-i b \lambda \text {ArcSin}(K[1])^n \left (\text {ArcSin}(K[1])^2\right )^{-n} \left ((-i \text {ArcSin}(K[1]))^n \text {Gamma}(n+1,i \text {ArcSin}(K[1]))-(i \text {ArcSin}(K[1]))^n \text {Gamma}(n+1,-i \text {ArcSin}(K[1]))\right )\right ) \left (-b \lambda \text {ArcSin}(K[1])^n+\lambda K[2] \text {ArcSin}(K[1])^n+a\right )}{a n \lambda (b+K[2])^2}\right )dK[1]-\frac {i \exp \left (a x-i b \lambda \text {ArcSin}(x)^n \left (\text {ArcSin}(x)^2\right )^{-n} \left ((-i \text {ArcSin}(x))^n \text {Gamma}(n+1,i \text {ArcSin}(x))-(i \text {ArcSin}(x))^n \text {Gamma}(n+1,-i \text {ArcSin}(x))\right )\right )}{a n \lambda (b+K[2])^2}\right )dK[2]=c_1,y(x)\right ] \]