Consider the problem $$f(x)y(x)''+\lambda y(x)=0$$ with $$y(0)=y(1)=0$$
In many cases it is not immediate finding the analytical solution of the differential equations because it depends on the coefficient $f(x)$. In the buckling problem it is sufficient evaluating the first value of $\lambda$ (the smallest eigenvalue, namely critical load). I would substitute f(x) with another similar function g(x) such that the differential equation can be solved. The new equation will be
$$g(x)y(x)''+\hat \lambda y(x)=0$$ with $$y(0)=y(1)=0$$
Question: Can I evaluate the maximum error among $\lambda$ and $\hat \lambda$ in terms of the relation between $f(x)$ and $g(x)$?