Final answer:
The given differential equation can be solved by separating it into a transient or decaying part (T(t)) and a steady state or permanent part (S(t)). The general form of T(t) can be found by assuming a solution of the form e^(rt) and solving for r. The steady state response, S(t), can be written as Acos(ωt) + Bsin(ωt), with A and B determined by the coefficients of the differential equation. The initial conditions can be applied by substituting the sum of T(t) and S(t) into the differential equation.
Step-by-step explanation:
The given differential equation is a second-order linear homogeneous differential equation with constant coefficients. The general solution to this type of differential equation is usually a sum of two parts: a transient or decaying part and a steady state or permanent part.
To detect the two parts, we can look at the behavior of the solutions over time. The transient part, T(t), represents the initial response of the system and decays over time. The steady state part, S(t), is the long-term behavior of the system after the transients have died out.
To find the general form of T(t), we can assume a solution of the form T(t) = e^(rt), where r is a constant. Substitute this into the differential equation and solve for r. The resulting roots will determine the form of T(t) based on the values of m, c, and k.
The steady state response, S(t), can be written as S(t) = Acos(ωt) + Bsin(ωt), where A and B are determined by the coefficients of the differential equation and ω is the angular frequency of the forcing function.
To apply the initial conditions, substitute y(t) = T(t) + S(t) into the differential equation and solve for A and B using the given initial conditions.