(Solved): (40 pts) Consider a causal continuous-time LTI system whose input \( x(t) \) and output \( y(t) \) ...

(40 pts) Consider a causal continuous-time LTI system whose input \( x(t) \) and output \( y(t) \) are related by the following differential equation: \[ \frac{d}{d t} y(t)+7 y(t)=2 \frac{d}{d t} x(t)+3 x(t) \] Find the Fourier series representation of the output \( y(t) \) for each of the following inputs: (Hint: \( \left.H(\omega)=\frac{Y(\omega)}{X(\omega)}\right) \) (c) \( x(t)=\sum_{n=-\infty}^{\infty}(-1)^{n} \delta(t-n) \) (d) \( x(t) \) is the periodic wave depicted below: