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Q2. Consider the electric circuit shown in Fig. 2. The differential equation that relates Figure 2: Q2 the input $x(t)$ and output $y(t)$ is $dt_{2}d_{2}yâ€‹+LC1â€‹y(t)=LC1â€‹x(t)$ (a). Find the characteristic equation for this circuit, and express the root(s) of the characteristic equation in terms of $L$ and $C$. (b). Determine the zero-input response given an initial capacitor voltage of 1 volt and an initial inductor current of zero amps. That is, find $y_{0}(t)$ given $v_{C}(0)=1ÂV$ and $i_{L}(0)=0$. (c). Plot $y_{0}(t)$ for $tâ‰¥0$. Does the zero-input response, which is caused solely by initial conditions, ever "die out"? (d). Determine the total response $y(t)$ to the input $x(t)=e_{âˆ’t}u(t)$. Assume an initial inductor current of $i_{L}(0)=0ÂA$, an initial capacitor voltage of $v_{C}(0)=1ÂV,L=1H$, and $C=1ÂF$.

The differential equation relating the input and output is given as,

Applying Laplace transforms