Internal problem ID [8951]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1374.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {y^{\prime \prime }-\frac {2 x \left (2 a -1\right ) y^{\prime }}{x^{2}-1}+\frac {\left (x^{2} \left (2 a \left (2 a -1\right )-v \left (v +1\right )\right )+2 a +v \left (v +1\right )\right ) y}{\left (x^{2}-1\right )^{2}}=0} \]
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 29
dsolve(diff(diff(y(x),x),x) = 2*x*(2*a-1)/(x^2-1)*diff(y(x),x)-(x^2*(2*a*(2*a-1)-v*(v+1))+2*a+v*(v+1))/(x^2-1)^2*y(x),y(x), singsol=all)
\[ y \left (x \right ) = c_{1} \left (x^{2}-1\right )^{a} \operatorname {LegendreP}\left (v , x\right )+c_{2} \left (x^{2}-1\right )^{a} \operatorname {LegendreQ}\left (v , x\right ) \]
✓ Solution by Mathematica
Time used: 0.016 (sec). Leaf size: 26
DSolve[y''[x] == -(((2*a + v*(1 + v) + (2*a*(-1 + 2*a) - v*(1 + v))*x^2)*y[x])/(-1 + x^2)^2) + (2*(-1 + 2*a)*x*y'[x])/(-1 + x^2),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \left (x^2-1\right )^a (c_1 \operatorname {LegendreP}(v,x)+c_2 \operatorname {LegendreQ}(v,x)) \\ \end{align*}