Internal problem ID [8708]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1128.
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 +\left (x f \relax (x )+2\right ) y^{\prime }+f \relax (x ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.037 (sec). Leaf size: 35
dsolve(x*diff(diff(y(x),x),x)+(x*f(x)+2)*diff(y(x),x)+f(x)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = \frac {c_{1}}{x}+\frac {c_{2} \left (\int {\mathrm e}^{\int \frac {-f \relax (x ) x -2}{x}d x} x^{2}d x \right )}{x} \]
✓ Solution by Mathematica
Time used: 0.016 (sec). Leaf size: 37
DSolve[f[x]*y[x] + (2 + x*f[x])*y'[x] + x*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {c_2 \int _1^x\exp \left (-\int _1^{K[2]}f(K[1])dK[1]\right )dK[2]+c_1}{x} \\ \end{align*}