Internal problem ID [8826]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1246.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _exact, _linear, _homogeneous]]
Solve \begin {gather*} \boxed {\left (x^{2}-1\right ) y^{\prime \prime }-2 \left (v -1\right ) x y^{\prime }-2 y v=0} \end {gather*}
✓ Solution by Maple
Time used: 0.013 (sec). Leaf size: 35
dsolve((x^2-1)*diff(diff(y(x),x),x)-2*(v-1)*x*diff(y(x),x)-2*v*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} \left (x^{2}-1\right )^{v}+c_{2} \left (x^{2}-1\right )^{v} x \hypergeom \left (\left [\frac {1}{2}, v +1\right ], \left [\frac {3}{2}\right ], x^{2}\right ) \]
✓ Solution by Mathematica
Time used: 0.011 (sec). Leaf size: 32
DSolve[-2*v*y[x] - 2*(-1 + v)*x*y'[x] + (-1 + x^2)*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \left (x^2-1\right )^{v/2} (c_1 P_v^v(x)+c_2 Q_v^v(x)) \\ \end{align*}