Maximum principle for approximated solutions to a boundary value problem

Question

Consider the interval $S = [t_0,t^*]$ and a function $f \in C^2(S)$. Consider the second-order ODE $$L_\varepsilon f(t)= \frac{1}{2}\mu^2t^2 f''(t)+\frac{\mu^2t^2}{t+t_0+\varepsilon}f'(t)+1,$$ where $t \in S$, $\mu>0$ and $\varepsilon >0$. Function $v_\varepsilon \in C^2(S)$ is solution of the following boundary value problem:$$L_\varepsilon v_\varepsilon(t) = 0,$$ $$v_\varepsilon(t_0) = \alpha_\varepsilon,$$ $$v_\varepsilon(t^*) = 0,$$ where $\alpha_\varepsilon > 0$. We also have a function $\hat{v}_\varepsilon \in C^2(S)$ such that $$L_\varepsilon \hat{v}_\varepsilon(t) = 0 + \mathcal{O}(\varepsilon^2),$$ $$\hat{v}_\varepsilon(t_0) = \alpha_\varepsilon + \mathcal{O}(\varepsilon^2),$$ $$\hat{v}_\varepsilon(t^*) = 0 + \mathcal{O}(\varepsilon^2).$$ I am trying to find an upper bound on $||v_\varepsilon-\hat{v}_\varepsilon||_{\infty}$. My idea is that an "approximate" maximum principle should hold, i.e. since $v_\varepsilon$ and $\hat{v}_\varepsilon$ are "close" at the boundary, they should be close in the interior of $S$ as well. Unfortunately I am far from being an expert in this field and I do not know of any result that could help me out. Any suggestion, hint would be greatly appreciated.