Question

Given that v(0) = 5 and dv(0)/dt = 10, solve ((d² v(t)) / (dt²)) + 5 ((d v(t)) / (dt)) + 6v = 30 e⁻ᵗ u(t).

Answer 1

To solve the given second-order linear homogeneous differential equation, we assume a solution of the form v(t) = e^rt and find the characteristic equation. The general solution is v(t) = C1e^-2t + C2e^-3t. The values of C1 and C2 can be determined using the initial conditions.

Step-by-step explanation:

The given differential equation is a second-order linear homogeneous differential equation. To solve it, we can assume a solution of the form v(t) = ert. Plugging this into the differential equation, we can find the characteristic equation:

r2 + 5r + 6 = 0

Factoring the equation as (r + 2)(r + 3) = 0, we find two distinct roots: r = -2 and r = -3. Therefore, the general solution to the differential equation is v(t) = C1e-2t + C2e-3t. To find the values of C1 and C2, we can use the initial conditions.