Internal problem ID [8800]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1220.
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 x^{2} f \relax (x ) y^{\prime }+\left (x^{2} \left (f^{\prime }\relax (x )+f \relax (x )^{2}+a \right )-v \left (v -1\right )\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 51
dsolve(x^2*diff(diff(y(x),x),x)+2*x^2*f(x)*diff(y(x),x)+(x^2*(diff(f(x),x)+f(x)^2+a)-v*(v-1))*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} {\mathrm e}^{-\frac {\left (\int 2 f \relax (x )d x \right )}{2}} \sqrt {x}\, \BesselJ \left (v -\frac {1}{2}, x \sqrt {a}\right )+c_{2} {\mathrm e}^{-\frac {\left (\int 2 f \relax (x )d x \right )}{2}} \sqrt {x}\, \BesselY \left (v -\frac {1}{2}, x \sqrt {a}\right ) \]
✓ Solution by Mathematica
Time used: 0.04 (sec). Leaf size: 62
DSolve[y[x]*((1 - v)*v + x^2*(a + f[x]^2 + Derivative[1][f][x])) + 2*x^2*f[x]*y'[x] + x^2*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \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 ) \exp \left (\int _1^x\left (\frac {1}{2 K[1]}-f(K[1])\right )dK[1]\right ) \\ \end{align*}