Find Io(t) of RLC circuit (from Electric Circuits, Nilsson )
The given RLC circuit is overdamped as:
$$ \frac{R}{2L} &gt; \frac{1}{\sqrt{LC}}$$
Hence the general solution is the sum of two decaying exponentials:
$$i(t)= K_1e^{xt} + K_2e^{yt}$$
where x,y are roots of the characteristic equation:
$$s^2+\frac{R}{L}s+\frac{1}{LC}=0$$
ie., 
$$s^2+1250s+250000 = 0$$
$$\implies (s+1000)(s+250)=0$$
Hence,
$$i(t)=K_1e^{-250t}+K_2e^{-1000t}$$

Now consider the capacitor,$$v_o(t) =\int{\frac{i(t)}{C}}$$
Hence \$v_o(t)\$ should be in the form:
$$v_o(t)=\frac{K_1}{-250C}e^{-250t}+\frac{K_2}{-250C}e^{-1000t}+K_3$$

Analysing the initial conditions will help you solve the constants ...