$1$ ODE $(40$ Pts$)$

A small object of mass $m$ is moving through a viscous fluid under the influence of gravity. The velocity $v(t)$ of the object is governed by the following differential equation:

$m\frac{(d)v}{(d)t}=mg-bv,$

where: $m$ is the mass of the object, $g$ is the acceleration due to gravity, $b$ is the damping coefficient due to viscous drag. Let $v(0)=0($the object starts at rest$).$ Please answer the following questions with detailed intermediate steps.

We want to compute the velocity history, $v(t),$ of the small object. Solve the equation numerically for $v(t)$ using the forward Euler method. Write the update equation for $v(t)$ and compute $v(t)$ at $t=0\mathrm{.}1,0\mathrm{.}2,$dots,$1\mathrm{.}0$ seconds. Use the following parameters:

$m=2kg,g=9\mathrm{.}8\frac{m}{s^2},b=1k\frac{g}{s},$

Time step $t=0\mathrm{.}1s.$

The coordinate history, $x(t),$ is also of interest. However, the current equation is a second order linear equation in term of $x$ :

$m\frac{d^{2}x}{(d)t^2}=mg-b\frac{(d)x}{(d)t}.$

Rewrite the differential equation in its first$-$order form. Apply the RK$4$ to compute $x(0\mathrm{.}2)$ assuming the initial condition is $x(0)=0.$

Discuss what is the maximum time step that the forward Euler remains stable and also given the stability region plot of RK$4,$ what is its maximum stable time step?