Consider the RLC circuit, where the initial current flowing through the circuit at time t $=0$ is I$=5,$ and the initial charge on the capacitor at time t$=0$ is Q$=2.$ The components have values of R$=100,$ L$=5$ H$,$ and C$=\frac{1}{450,500}$F$.$

a$.)$ Write a differential equation for Q$($t$),$ the charge across the capacitor, assuming the voltage source V$($t$)=0.($NOTE: I think the equation might be LQ$\text{'}\text{'}+$RQ$\text{'}+\frac{1}{C}$Q$=$V$($t$)$ for the charge Q$($t$),$ but I$\text{'}$m not sure$).$

b$.)$ Identify an appropriate solution technique, and solve the differential equation for Q$($t$).$ Show the process and all the steps in your work.