Internal problem ID [7737]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 157.
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 }+a \left (y^{2}-2 y x +1\right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 231
dsolve((x^2-1)*diff(y(x),x) + a*(y(x)^2-2*x*y(x)+1)=0,y(x), singsol=all)
\[ y \relax (x ) = \frac {8 c_{1} \left (x +1\right ) \left (\left (a -\frac {1}{2}\right ) x -\frac {a}{2}+\frac {1}{2}\right ) \HeunC \left (0, -2 a +1, 0, 0, a^{2}-a +\frac {1}{2}, \frac {2}{x +1}\right )-a \left (-\frac {x}{2}-\frac {1}{2}\right )^{-2 a +1} \left (x +1\right ) \HeunC \left (0, 2 a -1, 0, 0, a^{2}-a +\frac {1}{2}, \frac {2}{x +1}\right )-8 \left (\HeunCPrime \left (0, -2 a +1, 0, 0, a^{2}-a +\frac {1}{2}, \frac {2}{x +1}\right ) c_{1}-\frac {\left (-\frac {x}{2}-\frac {1}{2}\right )^{-2 a +1} \HeunCPrime \left (0, 2 a -1, 0, 0, a^{2}-a +\frac {1}{2}, \frac {2}{x +1}\right )}{4}\right ) \left (x -1\right )}{4 \left (x +1\right ) \left (\HeunC \left (0, -2 a +1, 0, 0, a^{2}-a +\frac {1}{2}, \frac {2}{x +1}\right ) c_{1}-\frac {\left (-\frac {x}{2}-\frac {1}{2}\right )^{-2 a +1} \HeunC \left (0, 2 a -1, 0, 0, a^{2}-a +\frac {1}{2}, \frac {2}{x +1}\right )}{4}\right ) a} \]
✓ Solution by Mathematica
Time used: 0.376 (sec). Leaf size: 47
DSolve[(x^2-1)*y'[x] + a*(y[x]^2-2*x*y[x]+1)==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {Q_a(x)+c_1 P_a(x)}{Q_{a-1}(x)+c_1 P_{a-1}(x)} \\ y(x)\to \frac {P_a(x)}{P_{a-1}(x)} \\ \end{align*}