Assuming the following differential equation on the interval $0<x<c$ with a rational function $f(x,c)$
$$\left(\frac{d^2}{dx^2}+f(x,c)\right)y(x,c)=0,$$
what kind of simplifications (if any) might occur for the solutions $y(x,c)$ when considering the case $0<c \ll 1$?
Also, when $c$ becomes small enough that the interval in questions consists only of two neighboring points, I assume that the differential equation above should reduce to a difference equation. Is this correct? And how exactly should that be managed?
EDIT:
If you would prefer an explicit example, please consider the ODE
$$\left(\frac{d^2}{dx^2}+\frac{c^2+x^2+x^4+x^{10}}{(1+x^2+x^{42})x^7}\right)y(x,c)=0$$
for the question above.
EDIT2:
Also, I think it is important to mention that at the left boundary $x=0$ we have Dirichlet boundary condition $y(0)=0$ and at the right boundary $x=c$ we have Neumann boundary condition $y'(c)=1$.