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.