Differential equation of circuit which contains R,L,C components
In order to solve this problem you may want to use mesh or nodal analysis. Personally I would use mesh, though both will work. You will have three loops, each representing one current (*i*1, *i*2, and *i*3). You need to solve for the current in each loop for each element in terms of voltage. Remember the following identities:

Current in a capacitor:



Current in an inductor:



From these you should be able to generate your differential equations. Then you can solve the system of equations.