Internal problem ID [10561]
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]
\[ \boxed {y^{\prime }-\lambda \arcsin \left (x \right )^{n} y^{2}-a y=a b -b^{2} \lambda \arcsin \left (x \right )^{n}} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 87
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)
\[ y \left (x \right ) = \frac {-b \lambda \left (\int \arcsin \left (x \right )^{n} {\mathrm e}^{-\left (\int \left (2 \arcsin \left (x \right )^{n} \lambda b -a \right )d x \right )}d x \right )-c_{1} b -{\mathrm e}^{-\left (\int \left (2 \arcsin \left (x \right )^{n} \lambda b -a \right )d x \right )}}{c_{1} +\lambda \left (\int \arcsin \left (x \right )^{n} {\mathrm e}^{-\left (\int \left (2 \arcsin \left (x \right )^{n} \lambda b -a \right )d x \right )}d x \right )} \]
✓ Solution by Mathematica
Time used: 7.093 (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]
\[ \text {Solve}\left [\int _1^x\frac {i \exp \left (a K[1]-i b \lambda \arcsin (K[1])^n \left (\arcsin (K[1])^2\right )^{-n} \left ((-i \arcsin (K[1]))^n \Gamma (n+1,i \arcsin (K[1]))-(i \arcsin (K[1]))^n \Gamma (n+1,-i \arcsin (K[1]))\right )\right ) \left (-b \lambda \arcsin (K[1])^n+\lambda y(x) \arcsin (K[1])^n+a\right )}{a n \lambda (b+y(x))}dK[1]+\int _1^{y(x)}\left (-\int _1^x\left (\frac {i \exp \left (a K[1]-i b \lambda \arcsin (K[1])^n \left (\arcsin (K[1])^2\right )^{-n} \left ((-i \arcsin (K[1]))^n \Gamma (n+1,i \arcsin (K[1]))-(i \arcsin (K[1]))^n \Gamma (n+1,-i \arcsin (K[1]))\right )\right ) \arcsin (K[1])^n}{a n (b+K[2])}-\frac {i \exp \left (a K[1]-i b \lambda \arcsin (K[1])^n \left (\arcsin (K[1])^2\right )^{-n} \left ((-i \arcsin (K[1]))^n \Gamma (n+1,i \arcsin (K[1]))-(i \arcsin (K[1]))^n \Gamma (n+1,-i \arcsin (K[1]))\right )\right ) \left (-b \lambda \arcsin (K[1])^n+\lambda K[2] \arcsin (K[1])^n+a\right )}{a n \lambda (b+K[2])^2}\right )dK[1]-\frac {i \exp \left (a x-i b \lambda \arcsin (x)^n \left (\arcsin (x)^2\right )^{-n} \left ((-i \arcsin (x))^n \Gamma (n+1,i \arcsin (x))-(i \arcsin (x))^n \Gamma (n+1,-i \arcsin (x))\right )\right )}{a n \lambda (b+K[2])^2}\right )dK[2]=c_1,y(x)\right ] \]