Question $3$:

consider the following differential equation

$2\frac{d^{3}y}{d{t}^3}+4\frac{d^{2}y}{d{t}^2}+6\frac{dy}{dt}+8y=10u(t)$

This is a third$-$order equation, so we can define three state variables as follows:

$x_1=y,x_2=y^{},x_3=y^{}$

and their derivatives are

$x_1^{}=x_2,x_2^{}=x_3,$ and $,x_3^{}=-4x_1-3x_2-2x_3+5u(t)$

Write the state$-$space form of the system based on this general form:

$x^{}(t)=Ax(t)+Bu(t)$

$y(t)=Cx(t)+Du(t)$

Calculate the Transfer function