Internal problem ID [8780]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1200.
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}+2 y^{\prime } x^{2}-v \left (v -1\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 33
dsolve(x^2*diff(diff(y(x),x),x)+2*x^2*diff(y(x),x)-v*(v-1)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} {\mathrm e}^{-x} \sqrt {x}\, \BesselI \left (v -\frac {1}{2}, x\right )+c_{2} {\mathrm e}^{-x} \sqrt {x}\, \BesselK \left (v -\frac {1}{2}, x\right ) \]
✓ Solution by Mathematica
Time used: 0.011 (sec). Leaf size: 45
DSolve[(1 - v)*v*y[x] + 2*x^2*y'[x] + x^2*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to e^{-x} \sqrt {x} \left (c_1 J_{v-\frac {1}{2}}(-i x)+c_2 Y_{v-\frac {1}{2}}(-i x)\right ) \\ \end{align*}