Solve Q3 10:14 PM BSelect textO: Select allCopy1. Assuming that both springs have spring constant k andthat there is a damping force proportional to velocity x' withdamping constant y, write down the differential equation thatdescribes the motion of the mass. Note that the expressionfor the force function depends on which of the three differentparts of its domain x lies within.tions x(0) = 2, x'(0) = 0. Find the solution x(t) of the initialvalue problem for 0<1< and si<5, where t andt; are the times of the first and second events. Describe thephysical situation corresponding to each of these two events.6. Consider the damped problem using the parameter valuesL = 1, m = 1, y = 0.1, and k = 1.2. The Heaviside, or unit step function, is defined by(a) Use a computer to draw a direction field of the corre-sponding dynamical system.(b) If you have access to computer software that is capableof solving event problems, solve for and plot the graphs of 298 / 683x(1) and x' (1) for the following sets of initial conditions:1, x2 c0, x < c.и(х — с) 3DUse the unit step function to express the differential equationin Problem 1 in a single line.i. x(0) = 2, x'(0) = 0ii. x(0) = 5, x'(0) = 03. Is the differential equation derived in Problems 1 and 2linear or nonlinear? Explain why.Give a physical explanation of why the limiting values of thetrajectories as t → o depend on the initial conditions.(c) Draw a phase portrait for the equivalent dynamical sys-4. In the case that the damping constant y > 0, find the crit-ical points of the differential equation in Problem 2 and dis-cuss their stability properties.5. Consider the case of an undamped problem (y = 0) usingthe parameter values L = 1, m = 1, k = 1 and initial condi-tem.7. Describe some other physical problems that could be for-mulated as event problems in differential equations.

Name: Solve Q3 10:14 PM BSelect textO: Select allCopy1. Assuming that both s
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Description: Solve Q3 10:14 PM BSelect textO: Select allCopy1. Assuming that both springs have spring constant k andthat there is a damping force proportional to velocity x' withdamping constant y, write down the differential equation thatdescribes the motion of