Second order ode with time dependent coefficients $\frac{d^{2} x}{dt^{2}}-f(t)\frac{d x}{dt}-2 f(t) g(t)=0$ I'm looking for a solution (general expression) for the following second order ODE:
\begin{equation}
\frac{d^{2} x}{dt^{2}}-f(t)\frac{d x}{dt}-2 f(t) g(t)=0
\end{equation}
where $f(t),g(t)$ are continuous and differentiable functions.

$$
x'' -f(t) \left[x' -2g(t)\right] = 0
$$
let $v = x' -2g(t)$ then we have
$$
v' +2g' = x''
$$
or
$$
v' +2g' -f(t) v = 0
$$
so a general solution could look like
$$
v(t)\mathrm{e}^{\int_0^t f(s)ds} = \int_0^t 2g'(s)\mathrm{e}^{\int_0^s f(\tau)d\tau} ds
$$
depending on if you can solve the integrals ...
$\textbf{edit:}$
$$
v(t) = \mathrm{e}^{-\int_0^t f(s)ds}\int_0^t 2g'(s)\mathrm{e}^{\int_0^s f(\tau)d\tau} ds = x' - 2g(t)
$$
so we have the final integral being
$$
x(t) = \int_0^t \left[2g(\tau) + \mathrm{e}^{-\int_0^\tau f(s)ds}\int_0^\tau 2g'(s)\mathrm{e}^{\int_0^s f(s')ds'} ds'\right]d\tau
$$
