Activity 1: The motion equation of damped mass-spring system as shown in Fig. 1 below is given as follows: m d²x dx +c+kx = f(t),t> 0 dt² dt where: x(t): displacement in meters. m: mass in kg. c: damping coefficient in kg/s. k: spring stiffness constant in N/m. t: time in seconds. f(t): external force applied on system in N. k f(t) m C Assume that: m = 2 kg, c = 3 kg/s, k = 20 N/m. Initially, assume f(t) = 0. X(t) Fig. 1: Damped mass-spring system. (1-a) Determine the general solution of x (t). (1-b) Given the initial conditions x (0) = 4, and x'(0) = 1, determine the particular solution of x(t). (1-c) Use Laplace transform to solve the given differential equation with the same initial conditions given in (1-b). (1-d) Critically evaluate how your mathematical solution behaves in transient and steady state regions. (1-e) Build a Simulink block model of the system, and generate the results of the differential equation. Show the model and the results in your report and specify the transient and steady state regions. Make sure you submit the simulation file (properly named). (1-f) Assume now that the external force f(t) equals 10 N, repeat all parts (1-a,b,c,d,e).

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Consider the differential equation: dy/dx = y(3 - y) 1) Use the slope field plotter to plot different solutions corresponding to A: (0,-2); B: (0,1); C: (0,3); D: (0,4). You can drag each point to the correct place if they're somewhere else on the plane, or click on the point and redefine. Capture a screenshot showing all the solution curves and insert below. f'(x)=y(3-y) Kin/5 -5 Density Length dy de -8 Step size 0.1 Ymas y (3-9) Input. ✔Solution A Solution B Solution C Solution D DCQ= AC: 2) To what value does y(x) approach, as x → ∞, for the solutions corresponding to B,C, and D? To what value does y(x) approach, as x →∞, for the solution corresponding to A? Why is it different from the others? Be specific and answer in terms of what you know about the slope field and its effect on the solution. - 3) Use wolframalpha.com or something similar to find the particular solution corresponding to each of the points A,B,C,D. How is the solution through Point C different from the other…

Consider the differential equation following is the slope of a line-segment 3 -1 0 07 - 2y = 3x. If sketching a slope field, which one of the at the point (x, y) = (1, 2)?