Laplace Transform (Voltage Transformation)

Question

So I was solving some circuits with the use of Voltage Transformation and Laplace Transform and I came up with this circuit. I'm not sure how to solve for IL and IC.

R3 and V1 are the voltage transformation of a 2H inductor while R4 and V2 are the voltage transformation of a 1/2 F capacitor.

By observation, I think it would be easier to choose nodal analysis. However, I'm not sure how to properly arrange the components to limit the number of nodes/equations needed.

^{simulate this circuit – Schematic created using CircuitLab}

Answer 1

To be solved by Laplace transforms, the entire circuit must be in the frequency domain. Therefore I2 must be represented by 2/(S+1). The 4 for V1 states that Il=2A at t=0+. When the topological graph is drawn for the circuit (opening current sources and shorting voltage sources), there is only one loop around which to write a mesh equation which is: (2/s)Ic+(2S)Il+Ir2+3Ir1=4-4/S; where Ir2 is the downward current in R2. Along with ancillary equations relating the various currents, we finally get: Il=(2S^2+5S+2)/(S+1)^3. Using partial fraction expansion and taking the inverse Laplace transform we get: i(t)=[(-1/2)t^2 + t + 2)]e^-t). As a check, evaluating at t=0+, we get il(0+)=2 which is expected.