Internal problem ID [8935]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1356.
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 {\left (2 x^{2}+1\right ) y^{\prime }}{x \left (x^{2}+1\right )}+\frac {\left (-v \left (v +1\right ) x^{2}-n^{2}\right ) y}{x^{2} \left (x^{2}+1\right )}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.026 (sec). Leaf size: 29
dsolve(diff(diff(y(x),x),x) = -(2*x^2+1)/x/(x^2+1)*diff(y(x),x)-(-v*(v+1)*x^2-n^2)/x^2/(x^2+1)*y(x),y(x), singsol=all)
\[ y \relax (x ) = c_{1} \LegendreP \left (v , n , \sqrt {x^{2}+1}\right )+c_{2} \LegendreQ \left (v , n , \sqrt {x^{2}+1}\right ) \]
✓ Solution by Mathematica
Time used: 0.163 (sec). Leaf size: 78
DSolve[y''[x] == -(((-n^2 - v*(1 + v)*x^2)*y[x])/(x^2*(1 + x^2))) - ((1 + 2*x^2)*y'[x])/(x*(1 + x^2)),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_1 x^{-n} \, _2F_1\left (\frac {1}{2} (-n-v),\frac {1}{2} (-n+v+1);1-n;-x^2\right )+c_2 x^n \, _2F_1\left (\frac {n-v}{2},\frac {1}{2} (n+v+1);n+1;-x^2\right ) \\ \end{align*}