Internal problem ID [3554]
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]
\[ \boxed {\left (-x^{2}+1\right ) y^{\prime }-n \left (y^{2}-2 y x +1\right )=0} \]
✓ Solution by Maple
Time used: 0.016 (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)
\[ y \left (x \right ) = \frac {8 \left (x +1\right ) c_{1} \left (x \left (n -\frac {1}{2}\right )-\frac {n}{2}+\frac {1}{2}\right ) \operatorname {HeunC}\left (0, -2 n +1, 0, 0, n^{2}-n +\frac {1}{2}, \frac {2}{x +1}\right )-n \left (-\frac {x}{2}-\frac {1}{2}\right )^{-2 n +1} \left (x +1\right ) \operatorname {HeunC}\left (0, 2 n -1, 0, 0, n^{2}-n +\frac {1}{2}, \frac {2}{x +1}\right )-8 \left (\operatorname {HeunCPrime}\left (0, -2 n +1, 0, 0, n^{2}-n +\frac {1}{2}, \frac {2}{x +1}\right ) c_{1} -\frac {\left (-\frac {x}{2}-\frac {1}{2}\right )^{-2 n +1} \operatorname {HeunCPrime}\left (0, 2 n -1, 0, 0, n^{2}-n +\frac {1}{2}, \frac {2}{x +1}\right )}{4}\right ) \left (x -1\right )}{4 n \left (x +1\right ) \left (\operatorname {HeunC}\left (0, -2 n +1, 0, 0, n^{2}-n +\frac {1}{2}, \frac {2}{x +1}\right ) c_{1} -\frac {\left (-\frac {x}{2}-\frac {1}{2}\right )^{-2 n +1} \operatorname {HeunC}\left (0, 2 n -1, 0, 0, n^{2}-n +\frac {1}{2}, \frac {2}{x +1}\right )}{4}\right )} \]
✓ Solution by Mathematica
Time used: 0.371 (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 {\operatorname {LegendreQ}(n,x)+c_1 \operatorname {LegendreP}(n,x)}{\operatorname {LegendreQ}(n-1,x)+c_1 \operatorname {LegendreP}(n-1,x)} y(x)\to \frac {\operatorname {LegendreP}(n,x)}{\operatorname {LegendreP}(n-1,x)} \end{align*}