Throughout Chapter 3 we have used the harmonic oscillator as an example. We have solved the second- order equation (Section 3.6) and its associated system of equations (Sections 3.2, 3.4, and 3.5) in a number of difference cases (i.e., two distinct nonzero real eigenvalues, complex eigenvalues, repeated eigenvalues, and a zero eigenvalue). Now it is time to summarize all that we have learned about this model (a) (3 points) Consider the harmonic oscillator with mass m=1, damping coefficient b = 5, and spring constant k = 4. For the values specified, write the second-order differential equation. Then find the general solution in scalar form of this second-order equation using the technique from Section 3.6. Classification of the oscillator (circle one): undamped underdamped overdamped critically damped (b) (1 point) Compute v(t). (Recall that y(t) = dy/dt!) (C) (1 point) Let Y(e) = (CE) be a vector-valued function. Using the results obtained in parts (a) and (b), write Y(t) in the following vector form Y(t) = kieOtVi+k2e At V2 where V1 and V2 are constant vectors and k and k2 are arbitrary constants (from part (a)). (d) (1 point) Now, convert the second-order equation you wrote in part (a) into a first-order system and write in matrix form. As usual, let v = dy dt. (e) (3 points) Compute the general solution of the first-order system from part (d). Type of the equilibrium point at the origin (circle one): sink source saddle spiral sink spiral source center (f) (1 point) Compare the results obtained in parts (c) and (e). What do you notice? Are you surprised by the results!? At this point, you may be wondering when you should use the scalar form of the general solution and when you should use the vector form. Discuss the advantages/disadvantages of each form