Internal problem ID [9642]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second
edition
Section: Chapter 1, section 1.2. Riccati Equation. 1.2.2. Equations Containing Power Functions
Problem number: 55.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, _Riccati]
Solve \begin {gather*} \boxed {\left (x^{2}-1\right ) y^{\prime }+\lambda \left (y^{2}-2 y x +1\right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 214
dsolve((x^2-1)*diff(y(x),x)+lambda*(y(x)^2-2*x*y(x)+1)=0,y(x), singsol=all)
\[ y \relax (x ) = \frac {\lambda \left (\HeunC \left (0, 2 \lambda +1, 0, 0, \lambda ^{2}+\lambda +\frac {1}{2}, \frac {2}{x +1}\right ) c_{1}-\left (-\frac {x}{2}-\frac {1}{2}\right )^{2 \lambda +1} \HeunC \left (0, -2 \lambda -1, 0, 0, \lambda ^{2}+\lambda +\frac {1}{2}, \frac {2}{x +1}\right )\right ) \left (x +1\right )}{2 \left (\left (\lambda +\frac {1}{2}\right ) x -\frac {\lambda }{2}-\frac {1}{2}\right ) c_{1} \left (x +1\right ) \HeunC \left (0, 2 \lambda +1, 0, 0, \lambda ^{2}+\lambda +\frac {1}{2}, \frac {2}{x +1}\right )-\lambda \left (-\frac {x}{2}-\frac {1}{2}\right )^{2 \lambda +1} \left (x +1\right ) \HeunC \left (0, -2 \lambda -1, 0, 0, \lambda ^{2}+\lambda +\frac {1}{2}, \frac {2}{x +1}\right )+2 \left (\HeunCPrime \left (0, 2 \lambda +1, 0, 0, \lambda ^{2}+\lambda +\frac {1}{2}, \frac {2}{x +1}\right ) c_{1}-\left (-\frac {x}{2}-\frac {1}{2}\right )^{2 \lambda +1} \HeunCPrime \left (0, -2 \lambda -1, 0, 0, \lambda ^{2}+\lambda +\frac {1}{2}, \frac {2}{x +1}\right )\right ) \left (x -1\right )} \]
✓ Solution by Mathematica
Time used: 0.355 (sec). Leaf size: 47
DSolve[(x^2-1)*y'[x]+\[Lambda]*(y[x]^2-2*x*y[x]+1)==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {Q_{\lambda }(x)+c_1 P_{\lambda }(x)}{Q_{\lambda -1}(x)+c_1 P_{\lambda -1}(x)} \\ y(x)\to \frac {P_{\lambda }(x)}{P_{\lambda -1}(x)} \\ \end{align*}