Internal problem ID [8954]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1375.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+\frac {2 x \left (n +1-2 a \right ) y^{\prime }}{x^{2}-1}+\frac {\left (4 a \,x^{2} \left (a -n \right )-\left (x^{2}-1\right ) \left (2 a +\left (v -n \right ) \left (v +n +1\right )\right )\right ) y}{\left (x^{2}-1\right )^{2}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.014 (sec). Leaf size: 39
dsolve(diff(diff(y(x),x),x) = -2*x/(x^2-1)*(n+1-2*a)*diff(y(x),x)-(4*a*x^2*(a-n)-(x^2-1)*(2*a+(v-n)*(v+n+1)))/(x^2-1)^2*y(x),y(x), singsol=all)
\[ y \relax (x ) = c_{1} \left (x^{2}-1\right )^{a -\frac {n}{2}} \LegendreP \left (v , n , x\right )+c_{2} \left (x^{2}-1\right )^{a -\frac {n}{2}} \LegendreQ \left (v , n , x\right ) \]
✓ Solution by Mathematica
Time used: 0.023 (sec). Leaf size: 34
DSolve[y''[x] == -(((4*a*(a - n)*x^2 - (2*a + (-n + v)*(1 + n + v))*(-1 + x^2))*y[x])/(-1 + x^2)^2) - (2*(1 - 2*a + n)*x*y'[x])/(-1 + x^2),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \left (x^2-1\right )^{a-\frac {n}{2}} (c_1 P_v^n(x)+c_2 Q_v^n(x)) \\ \end{align*}