Internal problem ID [9816]
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.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
Solve \begin {gather*} \boxed {y^{\prime }-y^{2}-\lambda x \arcsin \relax (x )^{n} y-\arcsin \relax (x )^{n} \lambda =0} \end {gather*}
✓ Solution by Maple
Time used: 0.006 (sec). Leaf size: 55
dsolve(diff(y(x),x)=y(x)^2+lambda*x*arcsin(x)^n*y(x)+lambda*arcsin(x)^n,y(x), singsol=all)
\[ y \relax (x ) = \frac {{\mathrm e}^{\int \frac {\arcsin \relax (x )^{n} \lambda \,x^{2}-2}{x}d x}}{c_{1}-\left (\int {\mathrm e}^{\int \frac {\arcsin \relax (x )^{n} \lambda \,x^{2}-2}{x}d x}d x \right )}-\frac {1}{x} \]
✓ Solution by Mathematica
Time used: 2.008 (sec). Leaf size: 171
DSolve[y'[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 {x+\frac {\exp \left (\lambda \left (-2^{-n-3}\right ) \text {ArcSin}(x)^n \left (\text {ArcSin}(x)^2\right )^{-n} \left ((-i \text {ArcSin}(x))^n \text {Gamma}(n+1,2 i \text {ArcSin}(x))+(i \text {ArcSin}(x))^n \text {Gamma}(n+1,-2 i \text {ArcSin}(x))\right )\right )}{\int _1^x\frac {\exp \left (-2^{-n-3} \lambda \text {ArcSin}(K[1])^n \left (\text {ArcSin}(K[1])^2\right )^{-n} \left (\text {Gamma}(n+1,2 i \text {ArcSin}(K[1])) (-i \text {ArcSin}(K[1]))^n+(i \text {ArcSin}(K[1]))^n \text {Gamma}(n+1,-2 i \text {ArcSin}(K[1]))\right )\right )}{K[1]^2}dK[1]+c_1}}{x^2} \\ y(x)\to -\frac {1}{x} \\ \end{align*}