Engineering Difference equation of simple RLC Circuit

[Eng67]

I have been having a difficult time understanding how to determine the input/output difference equations of a circuit. I believe I am good on series circuits but would like some feedback on Series/parallel determinations. Please look at included file and see if I am on the correct path.

 

[doodle]

Are you sure the answer you gave is correct?

I got
L d2(i_R)/dt2 + R d(i_R)/dt + (1/C) i_R = L d2(i)/dt2 + (1/C) i

or if you can render in Latex
L \frac{d^2 i_R}{dt^2} + R \frac{di_R}{dt} + \frac{i_R}{C} = L \frac{d^2 i}{dt^2} + \frac{i}{C}