Consider the parallel circuit in the figure with $~R = 20 ~\omega ,~~ C = \frac{1}{5}~F~$ and $~L = 5~H~$ with initial conditions $~iL=-1A~$ and $~Vc(0)= 4V~$ find the voltage $~v(t)~$.
So according to me to solve the problem I need $~V0=R1i+L\frac{di}{dt} +V~$ and $~i=Cdv/dt~$ obtaining from sustien $$ ~V0 = R1\left(C\frac{dv}{dt}\right)+L\frac{d}{dt}\left(C\frac{dv}{dt}\right) +V = R1CV'+LCV''+V= LCV''+R1CV'+V= r^2+ar+b=0~$$ getting an homogeneus differential equation and getting the roots but I'm not using the initial V and I.
Am I missing something?