I only remember a few steps in these kinds of problems...

BTW Z = Z_0 e^{-at} not just e^{-at}

Y = PI + CF = particular integral + complementary function I believe

You're given

Dy = cZ - gY where D = differentiation operator so

(D+g) Y = cZ

PI (or CF) is given by (D+g) Y = 0 i.e. PI = const * e^{-gt}

CF (or PI) is given by (D+g) Y = cZ = he^{-at} where h = cZ_0

Assume CF= ve^{-wt} then (D+g) Y = v(g-w)e^{-wt} , v w const

Eqauating powers of t, w = a so CF = ve^{-at}

v(g-a)e^{-at} = cZ_0e^{-at} so v = cZ_0/(g-a)

Now Y = PI + CF = ke^{-gt} + cZ_0/(g-a) e^{-at}

Almost there, hopefully someone more knowledgeable will correct my missteps here