(Solved): Problem 1 Consider the electric circuit shown in Fig. 1 Fig. 1 The differential equation that rel ...

Problem 1 Consider the electric circuit shown in Fig. 1 Fig. 1 The differential equation that relates the input voltage, \( v_{i}(t) \), to output voltage, \( v_{o}(t) \), is given by: \[ \left(C_{1}+C_{2}\right) \frac{d v_{o}(t)}{d t}+\frac{v_{o}(t)}{R}=C_{1} \frac{d v_{i}(t)}{d t} \] a. Using Impulse Matching Method, obtain the unit impulse response for the circuit. Assume \( C_{1}=C_{2}=1 F \) and \( R=0.5 \Omega \). b. Using Simplified Impulse Matching Method, find the unit impulse response of a system specified by the equation \[ \left(D^{2}+5 D+6\right) y(t)=(2 D+1) x(t) \] Problem 2 a. If \( y(t)=x_{1}(t) * x_{2}(t)=e^{-(t+3)} u(t+3) \), find \( z(t)=x_{1}(t+4) * x_{2}(t-5) \). b. If \( x_{1}(t)=8 e^{2 t} u(-t) \) and \( x_{2}(t)=0.25 \delta(t-4) \), find \( y(t)=x_{1}(t) * x_{2}(t) \). c. If \( x_{1}(t)=4 e^{-2 t} u(t) \) and \( x_{2}(t)=u(t) \), find \( y(t)=x_{1}(t) * x_{2}(t) \)