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MATH202–21S2 Assignment 2

        Your completed assignment should be handed in via the Dropbox on Learn. You may do the assignment on your own, or as a pair with one other student. If you do the assignment as a pair then you will both get the same mark. If you hand in a joint assignment, include both your names on it, but just one submission on Learn is required; if you are making the submission for both of you, then provide your assignment partner’s name in the submission comments on Learn, and if your partner is handing in the assignment on your behalf, please use the comments box to tell us that too!

Some marks will be given for the quality of presentation. To not lose marks here, your assign-ment must be clearly readable, written in complete sentences, with all figures clearly labelled and referenced in your text, and the actual questions answered! You should conclude with statements like: “The general solution to the differential equation given in part (b) is therefore .....”. It is not necessary to type your assignment.

A2.1. Consider the DE

Suppose that when r(t) = f(t) the solution of the DE is yf (t). Show that if r(t) = g(t) then the solution yg satisfies

Deduce that you can use the response of the system to one input to predict the response to another input, without knowing the constants in the DE!

A2.2. This question concerns a cantilevered beam of length l, that is clamped at x = 0, and free at the other end x = l. The equation for such a beam subjected to a vertical load w(x) at each point x along its length is

where y(x) is the deflection of the beam at x and k is a constant. You may assume that the weight of the beam is negligible, and that the load w(x) decreases linearly from w0 at x = 0 to zero at , and remains zero from  to l.

(a) Find a function f(x) in terms of w0 and the Heaviside function that extends the definition for w(x) to (0,∞). Note that, as the Laplace transform is an integral from 0 to ∞, you’ll need to extend the definition for w(x) to (0,∞) in order to apply the Laplace transform to w(x).

(b) Solve the initial value problem with

(c) Evaluate and in your solution, obtaining equations for a and b. Hence find a and b.

(d) Solve the initial value problem with the same initial conditions when the function f(x) is replaced by

that also extends w(x) to (0,∞). Is your solution the same as that obtained from the previous extension f(x) for x ∈ [0, l]? (The point of this question is less of a physical problem than a mathematical one, which is to determine the effects, if any, that the extension function and its domain have on the Laplace transform. Hence this part of the problem may not have a physical interpretation.)

A2.3. Consider the periodic functions that are defined by

and

(a) Calculate the Fourier series for f and g.

(b) Hence find the two polynomials that the two series

respectively converge to on the interval (0, 2π).

(c) Produce good quality graphs of the limit functions in (b), showing at least three periods.

(d) Use Maple or Matlab (or another alternative) to plot s4, s6, s8 on the same graph, where sN(t) is the truncation of the each series in (b) at the Nth term. Is there any overshoot?