$i_s(t)=\frac{V_{DC}}{R}\{1-e^{-\frac{t}{t_o}}\}u(t),$ where $t_o=\frac{L}{R}$

$[$Why does this solution make physical sense? Without performing any computations, can you do a quick hand$-$sketch of $i_s(t)$ versus $t?$ Why is $t_o=\frac{L}{R}$ called the "time$-$constant"?$]$

In this simulation lab we wish to solve $(2\mathrm{.}2),$ with applied voltage $v_s(t)=u(t)V_{DC},$ numerically ${?}^3$ using a computational mathematics technique $($called the variable step Runge$-$Kutta method$)$ that has been coded in Matlab as routine ODE$45.$ In order to use ODE$45$ we must first write $(2\mathrm{.}2)$ in so$-$called state space form ${?}^4$

$\frac{d{i}_s(t)}{dt}=-\frac{i_s(t)}{t_o}+\frac{v_s(t)}{L}$

Write a Matlab code that uses routine ODE$45$ to solve equation $(2\mathrm{.}5),$ and then plots $i_s(t)$ versus time t$.$ During the lab session you will be given values to use for $R,L$ and $V_{DC}.$