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In order to find \(H(s)\), we need to find \(X(s)\) and \(Y(s)\).

\begin{align*}

x(t) &= Ri + L\frac{di}{dt} + \frac{1}{C}\int i(t)dt\\

X(s) &= \mathcal{L}\bigg\{i + \frac{di}{dt} + \int i(t)dt\bigg\}\\

&= I(s) + sI(s) - I(0) + \frac{1}{s}I(s)\\

&= I(s)\bigg(1 + s + \frac{1}{s}\bigg)\\

y(t) &= \frac{1}{C}\int i(t)dt\\

&= \mathcal{L}\bigg\{\int i(t)dt\bigg\}\\

&= \frac{1}{s}I(s)\\

H(s) &= \frac{\frac{1}{s}}{1 + s + \frac{1}{s}}\\

&= \frac{1}{s^2 + s + 1}

\end{align*}

What is a causal system?

For convergece, \(\text{Re} \ \{s\} < -\frac{1}{2}\) since the inverse Laplace of H is

\[

\frac{2}{\sqrt{3}}e^{-\frac{1}{2}t}\sin\Big(\frac{\sqrt{3}}{2}t\Big).

\]