I need help to solve this problem, Im attaching the screenshots with the instructions for more information.

I General Advice: Module $4$

In Module $4,$ you will be asked to come up with an initial value problem. Remember that an initial value problem $($IVP$)$ consists of a differential equation

as well as initial conditions. The differential equation will be $m{y}^{\text{'}\text{'}}+ky=0$ in the undamped case and $m{y}^{\text{'}\text{'}}+c{y}^{\text{'}}+ky=0$ in the damped case as

previously described. As these are second$-$order differential equations, an initial value problem will include a value of $y(a)$ as well as $y^{\text{'}}(a).$ Remember

that that $y(t)$ represents the vertical distance between the spring's equilibrium position and the center of mass of the vehicle.

When designing your vehicle, you want to think about what kind of bumps your vehicle might encounter. This will depend on the type of environment

you will be using your vehicle in$.$ You will be using these bumps to design initial value problems. Do you expect your vehicle to have to go over $2-$foot

drops or only small bumps on a paved road? If your vehicle does go over a $2-$food drop, do you expect the wheels to stay in contact with the ground the

whole time? If not, how far from the equilibrium position will the wheels extend? How far will the wheels extend from the center of the vehicle? You will

use this to come up with your $y(a),$ the initial extension or compression of the wheels from their equilibrium position.

You will also need a value for $y^{\text{'}}(a).$ When you design your problems, remember that the differential equations we are using are modeling the vertical

movement of the vehicle. Therefore $y^{\text{'}}(a)$ is the initial vertical velocity not a horizontal velocity. If the vehicle is driving on a flat road and encounters a

dip, causing the wheels to drop, but stay in contact with the ground the whole time, then maybe you have an initial value given by $y^{\text{'}}(0)=0.$ But if the

dip in the road came while the vehicle was already headed down a hill, there will be a vertical component to the initial velocity. If your vehicle loses

contact with the ground for $2$ seconds $($which is probably not ideal for a family car but would be fine for a mountain bike$),$ then the vehicle may also

have picked up some initial velocity while it was in free$-$fall for two seconds so you likely want to calculate some non$-$zero $y^{\text{'}}(2)$ as the initial vertical

velocity which your vehicle has when the springs come in contact with the ground.