One way to model a vehicle is as a massive object that is connected by springs and shock absorbers to the wheels. Here is a simple model:

There is a mass m suspended by a spring with spring constant k and a oil-filled shock-absorber with viscous friction constant b over a wheel base. The wheels ride over a rough road that moves up and down. We will only model the vertical motion, with vertical displacement y. If you remember your basic physics you will be able to see that the balance-of-forces equation governing the motion in the y direction is just a second-order linear constant-coefficient ordinary differential equation:

md2y(t)dt2+bdy(t)dt+k(y(t)−Yroad(t))=0

To solve the differential equation provided, we can use the method of undetermined coefficients.

1. Start by assuming the solution has the form y(t) = Ae^(rt), where A and r are constants that we need to determine.

2. Differentiate y(t) twice to find its second derivative:
d^2y(t)/dt^2 = r^2Ae^(rt)

3. Substitute the solution and its derivatives back into the original differential equation:
m(r^2Ae^(rt)) + b(rAe^(rt)) + k(Ae^(rt) - Yroad(t)) = 0

4. Simplify the equation by factoring out common terms and combining like terms:
(Ae^(rt))(mr^2 + br + k) - kYroad(t) = 0

5. Since this equation must hold for all t, the term in parentheses must be equal to zero. Set it equal to zero and solve for r:
mr^2 + br + k = 0

The solutions to this quadratic equation for r give us the values of r that will satisfy the differential equation.

6. Once we have the values of r, we can determine the corresponding values of A using the initial conditions of the problem. This will give us the complete solution to the differential equation.

Note: The specific values of m, b, k, and Yroad(t) will need to be provided in order to obtain numerical values for the constants and solve the equation completely.