Internal problem ID [8733]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1153.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime } x^{2}+\left (a \,x^{2}-v \left (v -1\right )\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 33
dsolve(x^2*diff(diff(y(x),x),x)+(a*x^2-v*(v-1))*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} \sqrt {x}\, \BesselJ \left (v -\frac {1}{2}, x \sqrt {a}\right )+c_{2} \sqrt {x}\, \BesselY \left (v -\frac {1}{2}, x \sqrt {a}\right ) \]
✓ Solution by Mathematica
Time used: 0.014 (sec). Leaf size: 44
DSolve[((1 - v)*v + a*x^2)*y[x] + x^2*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \sqrt {x} \left (c_1 J_{v-\frac {1}{2}}\left (\sqrt {a} x\right )+c_2 Y_{v-\frac {1}{2}}\left (\sqrt {a} x\right )\right ) \\ \end{align*}