Problem Statement
The displacement $x(t)$ of a damped mechanical oscillator satisfies the second-order homogeneous differential equation:
$$\frac{d^2x}{dt^2} + 4\frac{dx}{dt} + 13x = 0$$
with initial conditions $x(0) = 2$ and $\left.\frac{dx}{dt}\right|_{t=0} = -1$.
**(a)** [2 marks] Write down and solve the characteristic auxiliary equation for this system.
**(b)** [4 marks] Find the particular solution for $x(t)$ satisfying the given initial conditions.
**(c)** [2 marks] State the type of damping exhibited by this system, and write down the quasi-frequency (angular frequency) of the oscillations.
Verified Solution & Marking Scheme
Part (a) [M1]
Characteristic equation: $r^2 + 4r + 13 = 0$
Part (a) [A1]
$r = \frac{-4 \pm \sqrt{16 - 52}}{2} = \frac{-4 \pm \sqrt{-36}}{2} = \frac{-4 \pm 6i}{2} = -2 \pm 3i$
Part (b) [M1]
General solution: $x(t) = e^{-2t}(A\cos 3t + B\sin 3t)$
Part (b) [A1]
$x(0) = e^0(A\cos 0 + B\sin 0) = A \implies A = 2$
Part (b) [M1]
Differentiate: $x'(t) = -2e^{-2t}(A\cos 3t + B\sin 3t) + e^{-2t}(-3A\sin 3t + 3B\cos 3t)$. At $t = 0$: $x'(0) = -2A + 3B = -1 \implies -2(2) + 3B = -1 \implies 3B = 3 \implies B = 1$
Part (b) [A1]
Particular solution: $x(t) = e^{-2t}(2\cos 3t + \sin 3t)$
Part (c) [A1]
Type of damping: Underdamped (oscillatory decay toward zero).
Part (c) [A1]
Quasi-frequency $\omega = 3\text{ rad/s}$