Problem $3$: $[26$pts$]$ Solve the following problems.

a$)[8$pts$]$ Consider a causal LTI system with frequency response given by

$H(j)=\frac{1}{j+5}$

For a particular input $x(t),$ this system is observed to produce the output

$y(t)=(e^{-5t}-e^{-3t})u(t)$

Determine $x(t).$

b$)[12$pts$]$ Consider the stable and causal LTI system described by the differential equation

$\frac{d^{2}y(t)}{d{t}^2}+6\frac{dy(t)}{dt}+8y(t)=2x(t)$

i$.$ Find the impulse response of this system.

ii$.$ What is the output $y(t)$ of this system if the input is $x(t)=t{e}^{-2t}u(t)?$

c$)[6$pts$]$ A causal and stable LTI system has the frequency response

$H(j)=\frac{j+4}{6-{}^2+5j}$

What is the differential equation relating the input $x(t)$ and the output $y(t)?$