Let's say I have a certain Sturm-Liouville problem with Dirichlet initial conditions of the form $$ (p(x) y'(x))'+q_c(x)y(x)=\lambda w(x) y(x), \quad y(a)=y(b)=0 $$ where the function $q_c$ is a smooth function depending on some parameter $c \in \mathbb{R}$.
Let's say also that I can explicitly compute the eigenvalues $\lambda_0 < \lambda_1< ...$ for the Sturm-Liouville problem $$ (p(x) y'(x))'+q_1(x)y(x)=\lambda w(x) y(x), \quad y(a)=y(b)=0 $$ this is, when $c=1$. It is known that under certain regularity assumptions (which we can assume) the eigenvalues depend continuously (and differentiably) on $q_c$. Is there a reasonable way of computing the eigenvalues for some other values of $c$? For example, obtaining a differential equation for each eigenvalue in terms of $c$ or something similar.