(Solved): A linear system is described by the circuit shown below. (a) Find the differential equation that r ...

A linear system is described by the circuit shown below. (a) Find the differential equation that relates the output \( y(t) \) (the voltage across the capacitor) and the input voltage \( x(t) \) in terms of \( R_{1}, R_{2}, L \) and \( C \). (b) For \( R_{1}=10 \Omega, R_{2}=2 \Omega, L=2 H \) and \( C=\frac{1}{4} F \) and given the initial conditions \( y(0)=2 \), \( \frac{d y(0)}{d t}=0 \), find the output \( y(t) \) for each of the following inputs, (i) \( x(t)=24 u(t) \) (ii) \( x(t)=\left(2 t^{2}+10 t+6\right) u(t) \) (iii) \( x(t)=10 e^{-5 t} u(t) \) (iv) \( x(t)=10 e^{-4 t} u(t) \) (v) \( x(t)=10 \cos \left(3 t+45^{\circ}\right) u(t) \)