## Simple RLC Circuit problem

Please help me to solve this RLC circuit problem. I am completely confused.If you give me the direct answer it would be much appreciated.
For the series RLC circuit in Figure, find the input/output
difference equation for

1.$$y(t)=v_{R}$$
2.$$Y(t)=i(t)$$
3.$$y(t)=v_{L}$$
4.$$y(t)=v_{C}$$

I have attached the Circuit diagram in a .jpg file.
Attached Thumbnails

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 Mentor You must show your own work in order for us to help you (PF homework forum rules). Would KCL or KVL be the best way to start?
 Hey first I tried taking the KVL around the loop something like $$-x(t) + v_c(t) + v_L (t) + v_R (t) = 0$$----(1) replaced v_L(t) with first order Ldi_L(t)/dt and make an equation for $$v_C(t)$$ and then as its in series I tried to write a function for $$i_L(t) = \frac {v_R (t)} R$$------(2) and for [ tex ] v_R(t)/R=C \frac {dv_c(t)} {dt} [/tex]----(3) Then tried sub (3) in (1) and got $$v_C(t) = x(t) - \frac {L} {R} dv_R(t)/dt - v + R(t)$$----(4) and then tried sub it i eqn 3. and didnt come up with a satisfactory result. Please help. Thanks

## Simple RLC Circuit problem

 Quote by Raihan Hey first I tried taking the KVL around the loop something like $$-x(t)+v_c(t)+v_L(t)+v_R(t)=0$$----(1) replaced v_L(t) with first order Ldi_L(t)/dt and make an equation for $$v_C(t)$$ and then as its in series I tried to write a function for $$i_L(t)=v_R(t)/R$$------(2) and for $$v_R(t)/R=Cdv_c(t)/dt$$----(3) Then tried sub (3) in (1) and got $$v_C(t)=x(t)-\frac {L} {R}dv_R(t)/dt-v+R(t)$$----(4) and then tried sub it i eqn 3. and didnt come up with a satisfactory result. Please help. Thanks
In series circuits you should always use $$v_C$$ as the independent variable (and $$i_L$$ in parallel circuits).
Since the current is the same for all elements, write $$v_L$$ and $$v_R$$ as functions of the current. Finally write the current as a function of $$v_C$$.
 In the second equation don't use the integral term. Keep it as $$V_C(t)$$. In the two other terms replace i by $$C\frac{dV_C}{dt}$$. You get a second order equation in $$V_C$$