61.14.2 problem 2

Internal problem ID [12235]
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 : 2
Date solved : Tuesday, January 28, 2025 at 01:31:47 AM
CAS classification : [_Riccati]

\begin{align*} y^{\prime }&=y^{2}+\lambda x \arcsin \left (x \right )^{n} y+\lambda \arcsin \left (x \right )^{n} \end{align*}

✓ Solution by Maple

Time used: 0.009 (sec). Leaf size: 78

dsolve(diff(y(x),x)=y(x)^2+lambda*x*arcsin(x)^n*y(x)+lambda*arcsin(x)^n,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 1.681 (sec). Leaf size: 256

DSolve[D[y[x],x]==y[x]^2+\[Lambda]*x*ArcSin[x]^n*y[x]+\[Lambda]*ArcSin[x]^n,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -\frac {\int _1^x\frac {\exp \left (-2^{-n-3} \lambda \arcsin (K[1])^n \left (\arcsin (K[1])^2\right )^{-n} \left (\Gamma (n+1,2 i \arcsin (K[1])) (-i \arcsin (K[1]))^n+(i \arcsin (K[1]))^n \Gamma (n+1,-2 i \arcsin (K[1]))\right )\right )}{K[1]^2}dK[1]+\frac {\exp \left (\lambda \left (-2^{-n-3}\right ) \arcsin (x)^n \left (\arcsin (x)^2\right )^{-n} \left ((-i \arcsin (x))^n \Gamma (n+1,2 i \arcsin (x))+(i \arcsin (x))^n \Gamma (n+1,-2 i \arcsin (x))\right )\right )}{x}+c_1}{x \left (\int _1^x\frac {\exp \left (-2^{-n-3} \lambda \arcsin (K[1])^n \left (\arcsin (K[1])^2\right )^{-n} \left (\Gamma (n+1,2 i \arcsin (K[1])) (-i \arcsin (K[1]))^n+(i \arcsin (K[1]))^n \Gamma (n+1,-2 i \arcsin (K[1]))\right )\right )}{K[1]^2}dK[1]+c_1\right )} \\ y(x)\to -\frac {1}{x} \\ \end{align*}