Suppose that I have a differential equation
$f''(x) = \mathcal{L}_1 (f)$
where $\mathcal{L}$ is an operator acting on $f$, with the solutions known in closed form.
Then I am trying to numerically solve the problem for $x\in[0,R]$
$f''(x) = \mathcal{L}_1(f) + g(x)\mathcal{L}_2(f)$
where now $g(x)$ is a known function and $\mathcal{L}_2$ is another operator acting on $f$. Given that
$\lim_{x\to0}g(x) =0$,
would it make sense to say that close to the origin $x=0$, the numerical solution would behave the same as the analytical solution (i.e. when $g(x)=0$)? In that case, the initial conditions near the origin can be given by the values of the analytical solutions.