Internal problem ID [8806]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1226.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {\left (x^{2}+1\right ) y^{\prime \prime }+2 x y^{\prime }-v \left (v -1\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 25
dsolve((x^2+1)*diff(diff(y(x),x),x)+2*x*diff(y(x),x)-v*(v-1)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} \LegendreP \left (-1+v , i x \right )+c_{2} \LegendreQ \left (-1+v , i x \right ) \]
✓ Solution by Mathematica
Time used: 0.007 (sec). Leaf size: 30
DSolve[(1 - v)*v*y[x] + 2*x*y'[x] + (1 + x^2)*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_1 P_{v-1}(i x)+c_2 Q_{v-1}(i x) \\ \end{align*}