ClearAll["Global`*"]  
;

(* Step 1: Before moving switch replace inductors with wires, caps with broken wires *)
step1=Solve[{V1==V0,V2==V1+2V0, V3==0,V0-0== 4 R1 I3},{V1,V2,V3,I1, I2,I3,I4}][[1]]//Quiet;
step1[[2]]
step1[[4]]

(* Step 2: After moving switch, replace with sources *)
(* I used the above results when creating the following equations: *)
step2=Solve[{V2==3V0,I3==(V0/(4 R1)),V1==V0, V1-V3==4 R1 I3,V2-0==3 R1 I4, I5==-I3-I4,I3==-I2},{V1,V2,V3,I1, I2,I3,I4,I5}][[1]]//Quiet;step2[[3]]
step2[[7]]

ClearAll["Global`*"]  
(* Step 3: After moving switch replace with wires *)
step3=Solve[{V2-V1==2V0,V2-V1==(2R1+R1+4R1)I4,V2-0==(2R1+R1)I4,I3==-I4},{V1,V2,V3,I1, I2,I3,I4,I5}][[1]]//Quiet;step3[[2]]
step3[[3]]

ClearAll["Global`*"]  
(* Step 4: Full circuit after moving switch *)
(* still using just "Solve"... I'm trying to not just solve the ODE, but find the ODE! *)
(* notice that I'm using V2p to mean V2', etc. *)
step4=Solve[{V1-V3==4R1 I3,V2-V4==2 R1 I4, V4-0==R1 I4,V2==2V0+V1,V3-0==L I3p,I5==C V2p,I3+I4+I5==0},{V1,V2,V3,V4,I3,I4,I5}][[1]]//Quiet;
step4[[5]]
step4[[2]]
step4[[4]]

ClearAll["Global`*"]  
(* Step 5: Find ODE? I don't know how to do this in Mathematica *)

ClearAll["Global`*"]  
(* Step 6: Solve ODE. Here, I'm just using DSolve instead of Solve, and doing the same
equations as step 4. Of course, I also added time dependence explicity. 
This gave bad results if I kept symbols for R, C, V0, etc. So I made up some numbers.
Also, this wouldn't even run if I used any decimals for the givens!
Notice there are 3 boundary/initial conditions! *)
V0=10;R1=20;L1=3;C1=(2/1000);
step6=DSolve[{V1[t]-V3[t]==4R1 I3[t],V2[t]-V4[t]==2 R1 I4[t], V4[t]-0==R1 I4[t],V2[t]==2V0+V1[t],V3[t]-0==L1 I3'[t],I5[t]==C1 V2'[t],I3[t]+I4[t]+I5[t]==0,V2[∞]== (60/7),V2[0]==3V0,I3[0]== (V0/(4 R1))},{V1[t],V2[t],V3[t],V4[t],I3[t],I4[t],I5[t]},t]//Flatten//Quiet;
V2[t]=1.0V2[t]/.step6//FullSimplify
Plot[V2[t],{t,0,10*R1*C1},PlotRange-> {0,30}]