Show that the equation

$$(z-1) w^{\prime \prime}-z w^{\prime}+(4-2 z) w=0$$

has solutions of the form $w(z)=\int_{\gamma} \exp (z t) f(t) d t$, where

$$f(t)=\frac{\exp (-t)}{(t-a)(t-b)^{2}}$$

and the contour $\gamma$ is any closed curve in the complex plane, where $a$ and $b$ are real constants which should be determined.

Use this to find the general solution, evaluating the integrals explicitly.