Problem $3$ Consider the following ODE$-$IVP:

$\frac{dy}{dt}=y(-2-5y-y^2)+0\mathrm{.}5cos(t),y(0)=3,$tin$[0,5]$

that represents a CSTR with a fluctuating feed stream and a second$-$order reaction.

a$)$ Write a MATLAB program that solves this problem numerically using RK$4.$ This function

should take as input the initial condition $(t_0,y_0),$ stepsize $h,$ and the final time $t_f,$ and

automatically generate a labeled plot of $y(t).$ Test this function for $h=0\mathrm{.}01$ and report your

results.

b$)$ Write a function that applies linear stability analysis to determine the maximum stepsize

required for stability at the current time $(t,y(t)).$ Using this function, determine the maximum

stepsize $h^{**}$ so that the system is stable at the first step.

c$)$ Using your code from Part $($a$),$ write a program that generates two plots: $(1)$ a plot of the

maximum stepsize versus $t,$ and $(2)$ a plot of the numerical solution $y$ versus $t$ for $h=0\mathrm{.}02,$

$h=h^{**},$ and $h=h^{**}+0\mathrm{.}035($on a single plot$).$ For full credit, don't forget to comment on

your findings.