Problem $2\mathrm{.}2($Numerical$)$ We will work on Problem $2\mathrm{.}1$ again, but this time do so computationally.

$($a$)$ Simulate the logistic equation with an ODE solver. Here, we set $r=0\mathrm{.}1,B=100,$ and

$N(t=0)=N_0=1$

$($b$)$ With the time trajectory of $N(t),$ you can now compute $\frac{d^{2}N}{d{t}^2}$ numerically using the following formula $($think

why; you don't need to derive this$)$

$\frac{d^{2}N}{d{t}^2}=\underset{t0}{lim}\frac{N(t+t)+N(t-t)-2N(t)}{t^2}$

Use the above formula to compute $\frac{d^{2}N}{d{t}^2}$ as the function of $t$ using the time trajectory from $($a$).$ Plot $\frac{d^{2}N}{d{t}^2}$ as the

function of $t.$

Previous logistic $2\mathrm{.}1$ equations:

dN$/$dt $=$ f$($N$)=$rN$(1-$N$/$B$)$

N$($t$)=($NB$)/($N$+($B$-$N$)$e$^-$rt$)$