Please see pdf attachment for question a) and provide me with advices, hints, answers, anything....

I have trying to prove the equation for 2 hours now, and I still dont know how.

Please help me. Thank you!

 

This is basically a nodal analysis problem.

On the circuit mark 3 currents

- the inductor current (also in series with RL and Rs) [i1(t)]
- the current in Rc [i2(t)]
- the current in C [i3(t)]

i1(t)=i2(t)+i3(t)

The current i1(t) in inductor L is related to the difference between Vs and Vc

Vs(t)-Vo(t)=(Rs+RL)*i1(t) + L*(di1(t)/dt)

The current in Rc, i2(t) =Vo(t)/Rc

The current in C, i3(t)=C*(dVo(t)/dt)=C*Vo'

Hopefully this will give you a start.

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