The dynamic response of a chemical process moving from one steady$-$state

condition to another can be represented by the following $2^{nd}$ order ordinary

differential equation.

$\frac{d^{2}y}{d{t}^2}+4\frac{dy}{dt}+3y=e^{-t},y(0)=1,y^{\text{'}}(0)=0$

$($a$)$ Briefly explain the method to be followed when solving differential

equations by using the Laplace transform technique.

$($b$)$ Use Laplace transform techniques to solve the differential equation.

$($c$)$ Sketch a graph of the expected dynamic response of the system over time.

Note that $y^{\text{'}}(0)$ represents the derivative of $y$ at time $t=0.$