Internal problem ID [9021]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1442.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+\frac {x y^{\prime }}{f \relax (x )}-\frac {y}{f \relax (x )}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.017 (sec). Leaf size: 31
dsolve(diff(diff(y(x),x),x) = -x/f(x)*diff(y(x),x)+1/f(x)*y(x),y(x), singsol=all)
\[ y \relax (x ) = c_{1} \left (\int {\mathrm e}^{\int \frac {-2-\frac {x^{2}}{f \relax (x )}}{x}d x}d x \right ) x +x c_{2} \]
✓ Solution by Mathematica
Time used: 0.039 (sec). Leaf size: 45
DSolve[y''[x] == y[x]/f[x] - (x*y'[x])/f[x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to x \left (c_2 \int _1^x\frac {\exp \left (-\int _1^{K[2]}\frac {K[1]}{f(K[1])}dK[1]\right )}{K[2]^2}dK[2]+c_1\right ) \\ \end{align*}