Internal problem ID [9821]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second
edition
Section: Chapter 1, section 1.2. Riccati Equation. subsection 1.2.7-2. Equations containing
arccosine.
Problem number: 11.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
\[ \boxed {y^{\prime }-y^{2}-\lambda x \arccos \left (x \right )^{n} y-\arccos \left (x \right )^{n} \lambda =0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 55
dsolve(diff(y(x),x)=y(x)^2+lambda*x*arccos(x)^n*y(x)+lambda*arccos(x)^n,y(x), singsol=all)
\[ y \left (x \right ) = \frac {{\mathrm e}^{\int \frac {\arccos \left (x \right )^{n} \lambda \,x^{2}-2}{x}d x}}{c_{1} -\left (\int {\mathrm e}^{\int \frac {\arccos \left (x \right )^{n} \lambda \,x^{2}-2}{x}d x}d x \right )}-\frac {1}{x} \]
✓ Solution by Mathematica
Time used: 3.067 (sec). Leaf size: 169
DSolve[y'[x]==y[x]^2+\[Lambda]*x*ArcCos[x]^n*y[x]+\[Lambda]*ArcCos[x]^n,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {x+\frac {\exp \left (\lambda 2^{-n-3} \arccos (x)^n \left (\arccos (x)^2\right )^{-n} \left ((-i \arccos (x))^n \Gamma (n+1,2 i \arccos (x))+(i \arccos (x))^n \Gamma (n+1,-2 i \arccos (x))\right )\right )}{\int _1^x\frac {\exp \left (2^{-n-3} \lambda \arccos (K[1])^n \left (\arccos (K[1])^2\right )^{-n} \left (\Gamma (n+1,2 i \arccos (K[1])) (-i \arccos (K[1]))^n+(i \arccos (K[1]))^n \Gamma (n+1,-2 i \arccos (K[1]))\right )\right )}{K[1]^2}dK[1]+c_1}}{x^2} \\ y(x)\to -\frac {1}{x} \\ \end{align*}