Question (1) [Part (a), (b) i-ii are revision from first year Calculus and thus have no allocated marks. You are expected to do these to gain the marks from the rest of the question, since these prerequisite skills underpin Nonlinear Systems in Topic 2. Check your own work on these parts later when the solutions and feedback are made available on LMS.] From first year Calculus, you learned that the oscillations of a mass-spring-damper system is governed by a single 2nd order ODE with constant coefficients for x(t): Where Ax + Bx + Cx = 0, dx(t) dt = x. (a) Express the system of coupled first order ODEs (1) dx dt = ax + by, dy = cx + dy (2) dt in the form of equation (1) for x(t). Confirm that you get the same ODE for y(t). (b) Consider the system in Exercise 8 pg 2.25 of Notes. i. Find the corresponding second order ODE in the form (1) and solve it using the method you learned in Calculus 2. ii. Classify the oscillation into one of the following types: underdamped, critically damped, overdamped. iii. Use MATLAB to plot the oscillatory displacement x(t) versus t from part i. c) Consider each of the following four types of oscillations: A. underdamped B. critically damped C. overdamped D. simple harmonic motion Identify the stability and type of critical point (0,0) that represents each type of oscillation and match it the relevant dynamical phase portrait(s) below. 1 3 4 * 5