Small parameter expansion of solution to ODE

Question

I'm working with an ODE of the form \begin{equation} \begin{split} C'(z) &= (z^2 + \epsilon^2)C'' \\ C(a) &= b \\ C(1) &= 1 \\ \end{split} \end{equation} where $0 < a,b < 1$, and $\epsilon$ is a small parameter. I know that the ODE can be solved exactly, but I want to exploit the size of $\epsilon$ to derive a more meaningful approximate result. Does an expansion of the form \begin{equation} C(z;\epsilon) = C^0(z)+\epsilon^2 C^1(z)+\cdots \end{equation} exist, where $C^{0}(z)$ satisfies the inhomogeneous boundary conditions? I tried working this out, but I'm not confident that such a simple expansion exists, since $\epsilon$ multiplies the highest order derivative. Is there are alternate way of approaching this problem?