M133a Due: Ex. 5 Spring 2017 (01) 04/12/17, R.K. Dodd (02) 04/13/17. Attempt all the questions. All problems must have work attached on the scratch sheets. NAME: 1. Find the solutions to the following IVPs: y 00 + 4y 0 + 3y = e t , (a) (b) (c) y 00 + y = sin x, y 00 + y 0 + y = 3x2 y(0) = 1, y 0 (0) = 2, y() = 0, y 0 () = 0. 2x + 1, y(0) = 1, y 0 (0) = 0. Ans. 2. For the following equations determine whether the method of undetermined coefficients can be used to obtain the particular solution. If possible write out the form of the particular solution. (a) y 00 +2y 0 y = x 1 sin x, (b) y 00 +y = x3 cos x+xe5x , (c) y (5) +y (2) +y = tan x. Ans. 3. Determine whether the following dierential operators are linear or nonlinear. (a) R[y] := y 00 y 0 +x2 y, (b) S[y] := y 00 +2y 0 sin x+6y, (c) T [y] := y 000 +y 2 . Ans. 4. Establish whether each of the following sets of functions are linearly independent or linearly dependent, (a) {sin 2x, e3x , x2 }, (b) {cos 3x, sin 2x, tan x}, (c) {x3 , x5 , cos x, x3 5x5 }, (d) {1, tan x, sec2 x}. Ans. 5. Suppose that the populations of two dierent animals P and Q in a given habitat are respectively p(t) and q(t). The Q animals have a growth rate which is proportional to their population, because the habitat has a plentiful supply of 1 their food. The P animals survive by preying only upon the Q animals and will die out exponentially if the Q population is very small. Assume that the rate at which the P population grows is proportional to the product of both their populations at any given time. Show that a possible model for this situation is provided by the equations dq = aq dt dp = dt bpq, cp + dpq, () where a, b, c and d are positive constants. The constant solutions of these equations are called equilibrium solutions. Thus for example if q = q , p = p is an equilibrium solution of the system then dq /dt = 0, dp /dt = 0. (a) Find the possible equilibrium solutions for the animal populations. (b) Recall that the average of a function f (t) over a time T is defined by 1 T Z T f (t)dt. 0 It can be shown that the populations of the animals vary periodically, that is there exists a time T such that p(t + T ) = p(t) and q(t + T ) = q(t). Use the equations () to show that the average values of the P, Q populations over a period T is one of the possible equilibrium solutions for the P and Q populations. Note: You do not need to know T to be able to do this problem. Ans. 2