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Given the differential equation $L \ u = f$, with $$L \ u = a_2(t) \frac{d^2u}{dt^2}+a_1(t)\frac{du}{dt}+a_0(t)u$$ with $a_i(t)$ sufficiently smooth and $a_2(t) \neq 0$ for every $t$. Suppose that $u_0$ is a solution to the homogenenous equation $L \ u=0$, with $u_0(t)\neq0$ for every $t$. How can I then use the variation of parameters method by Lagrange to find the solution $u =u_0v$ ? And how can I find the general solution if the equation isn't homogeneous?

Update: I've substituted $u=u_0v$ into the equation and found a new $2^{nd}$ order differential equation of the form $$f = \frac{d^2v}{dt^2}(a_2(t)u_0)+\frac{dv}{dt}(2a_2(t)u_0'+a_1(t)u_0)+va_0(t)u_0$$ which I'm supposed to solve for $w$ using $w=v'$. But I don't seem to be able to proceed as I'm not sure how to solve the system $$\begin{equation} \begin{split} w =&\ v'\\ w'a_2(t)u_0+w(2a_2(t)u_0'+a_1(t)u_0)+v(a_0(t)u_0)=&\ f\ \\ \end{split} \end{equation}$$

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