Problem $1.$ Consider the ordinary differential

equation $($ODE$)$:

$x^{}+ax=3a,$

where $a$ is constant, with the initial condition $($IC$)$:

$x(0)=2.$

a$)$ Determine the type of this ODE.

b$)$ Calculate the solution $x(t)$ analytically.

c$)$ For $a=1,$ calculate the relaxation time and

evaluate $\underset{t}{lim}x(t)$

d$)$ Use Matlab $($or other preferred software$)$ to plot the

analytical solution $x(t)$ for $a=-1,a=0,$ and $a=$

$1$ over the interval tin$[0,2].$

e$)$ Calculate the solution numerically in Matlab $($e$.$g$.$

via ode$45),$ plot it$,$ and compare it to the analytical

solution $($for both $a=-1,a=0,$ and $a=1).$

f$)$ Include your Matlab codes in your submission.