Internal problem ID [3046]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 11
Problem number: 298.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, _Riccati]
Solve \begin {gather*} \boxed {\left (1-x^{2}\right ) y^{\prime }-n \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) = n*(1-2*x*y(x)+y(x)^2),y(x), singsol=all)
\[ y \relax (x ) = \frac {8 c_{1} \left (x +1\right ) \left (\left (n -\frac {1}{2}\right ) x -\frac {n}{2}+\frac {1}{2}\right ) \HeunC \left (0, -2 n +1, 0, 0, n^{2}-n +\frac {1}{2}, \frac {2}{x +1}\right )-n \left (-\frac {x}{2}-\frac {1}{2}\right )^{-2 n +1} \left (x +1\right ) \HeunC \left (0, 2 n -1, 0, 0, n^{2}-n +\frac {1}{2}, \frac {2}{x +1}\right )-8 \left (x -1\right ) \left (\HeunCPrime \left (0, -2 n +1, 0, 0, n^{2}-n +\frac {1}{2}, \frac {2}{x +1}\right ) c_{1}-\frac {\left (-\frac {x}{2}-\frac {1}{2}\right )^{-2 n +1} \HeunCPrime \left (0, 2 n -1, 0, 0, n^{2}-n +\frac {1}{2}, \frac {2}{x +1}\right )}{4}\right )}{4 \left (x +1\right ) \left (\HeunC \left (0, -2 n +1, 0, 0, n^{2}-n +\frac {1}{2}, \frac {2}{x +1}\right ) c_{1}-\frac {\left (-\frac {x}{2}-\frac {1}{2}\right )^{-2 n +1} \HeunC \left (0, 2 n -1, 0, 0, n^{2}-n +\frac {1}{2}, \frac {2}{x +1}\right )}{4}\right ) n} \]
✓ Solution by Mathematica
Time used: 0.354 (sec). Leaf size: 47
DSolve[(1-x^2)y'[x]==n(1-2 x y[x]+y[x]^2),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {Q_n(x)+c_1 P_n(x)}{Q_{n-1}(x)+c_1 P_{n-1}(x)} \\ y(x)\to \frac {P_n(x)}{P_{n-1}(x)} \\ \end{align*}