Internal problem ID [8854]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1274.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {4 y^{\prime \prime } x^{2}+4 x y^{\prime }+\left (-v^{2}+x \right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 19
dsolve(4*x^2*diff(diff(y(x),x),x)+4*x*diff(y(x),x)+(-v^2+x)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} \BesselJ \left (v , \sqrt {x}\right )+c_{2} \BesselY \left (v , \sqrt {x}\right ) \]
✓ Solution by Mathematica
Time used: 0.021 (sec). Leaf size: 38
DSolve[(-v^2 + x)*y[x] + 4*x*y'[x] + 4*x^2*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_1 \Gamma (1-v) J_{-v}\left (\sqrt {x}\right )+c_2 \Gamma (v+1) J_v\left (\sqrt {x}\right ) \\ \end{align*}