� � � �� 4. Linear Systems 4A. Review of Matrices ⎝ � ⎞ ⎠ 1 0 −1 ⎞ ⎠ 2 0 1 1 0 0 4A-1. Verify that � 0 2 1 � = . 1 −1 −2 3 −2 −6 −1 0 2 ⎞ ⎠ ⎞ ⎠ 1 2 0 −1 4A-2. If A = and B = , show that AB �= BA. 3 −1 2 1 ⎞ ⎠ 4A-3. Calculate A−1 if A = 2 2 , and check your answer by showing that AA−1 = I 3 2 and A−1A = I. 4A-4. Verify the formula given in Notes LS.1 for the inverse of a 2 × 2 matrix. ⎞ ⎠ 0 1 4A-5. Let A = . Find A3 (= A · A · A).1 1 4A-6. For what value of c will the vectors x1 = (1, 2, c), x2 = (−1, 0, 1), and x3 = (2, 3, 0) be linearly dependent? For this value, find by trial and error (or otherwise) a linear relation connecting them, i.e., one of the form c1x1 + c2x2 + c3x3 = 0 4B. General Systems; Elimination; Using Matrices 4B-1. Write the following equations as equivalent first-order systems: d2x dx y�� − x2a) + 5 + tx2 = 0 b) y� + (1 − x2)y = sin x dt2 dt 4B-2. Write the IVP y(3) + p(t)y �� + q(t)y + r(t)y = 0, y(0) = y0, y (0) = y0, y ��(0) = y0 as an equivalent IVP for a system of three first-order linear ODE’s. Write this system both as three separate equations, and in matrix form. ⎞ ⎠ ⎞ ⎠ 1 1 x 4B-3. Write out x� = x, x = as a system of two first-order equations. 4 1 y a) Eliminate y so as to obtain a single second-order equation for x. b) Take the second-order equation and write it as an equivalent first-order system. This isn’t the system you started with, but show a change of variables converts one system into the other. 4B-4. For the system x� = 4x − y, y� = 2x + y, 2ta) using matrix notation, verify that x = e3t , y = e3t and x = e , y = 2e2t are solutions; b) verify that they form a fundamental set of solutions — i.e., that they are linearly independent; 1� 2 18.03 EXERCISES c) write the general solution to the system in terms of two arbitrary constants c1 and c2; write it both in vector form, and in the form x = . . . , y = . . . . ⎞ ⎠ 1 3 4B-5. For the system x� = Ax , where A = 3 1, ⎞ ⎠ ⎞ ⎠ 1 −2ta) show that x1 =1 e4t and x2 = e form a fundamental set of solutions 1 −1 (i.e., they are linearly independent and solutions); ⎞ ⎠ 5 b) solve the IVP: x� = Ax, x(0) = . 1 ⎞ ⎠ 1 1 4B-6. Solve the system x� = x in two ways: 0 1 a) Solve the second equation, substitute for y into the first equation, and solve it. b) Eliminate y by solving the first equation for y, then substitute into the second equation, getting a second order equation for x. Solve it, and then find y from the first equation. Do your two methods give the same answer? 4B-7. Suppose a radioactive substance R decays into a second one S which then also decays. Let x and y represent the amounts of R and S present at time t, respectively. a) Show that the physical system is modeled by a system of equations ⎞ ⎠ ⎞ ⎠ −a 0 x x = Ax, where A = , x = , a, b constants. a −b y b) Solve this sytem by either method of elimination described in 4B-6. c) Find the amounts present at time t if initially only R is present, in the amount x0. Remark. The method of elimination which was suggested in some of the preceding problems (4B-3,6,7; book section 5.2) is always available. Other than in these exercises, we will not discuss it much, as it does not give insights into systems the way the methods will decribe later do. Warning. Elimination sometimes produces extraneous solutions — extra “solutions” that do not actually solve the original system. Expect this to happen when you have to differentiate both equations to do the elimination. (Note that you also get extraneous solutions when doing elimination in ordinary algebra, too.) If you get more independent solutions than the order of the system, they must be tested to see if they actually solve the original system. (The order of a system of ODE’s is the sum of the orders of each of the ODE’s in it.) 4C. Eigenvalues and Eigenvectors ⎝ � ⎞ ⎠ ⎞ ⎠ 1 −1 0 −3 4 4 −3 4C-1. Solve x� = Ax, if A is: a) b) c) � 1 2 1 �. −2 3 8 −6 −2 1 −1 ( First find the eigenvalues and associated eigenvectors, and from these construct the normal modes and then the general solution.)SECTION 4. LINEAR SYSTEMS 3 4C-2. Prove that m = 0 is an eigenvalue of the n × n constant matrix A if and only if A is a singular matrix (detA = 0). (You can use the characteristic equation, or you can use the definition of eigenvalue.) 4C-3. Suppose a 3 × 3 matrix is upper triangular. (This means it has the form below, where � indicates an arbitrary numerical entry.) ⎝ � a � � A = � 0 b � � 0 0 c Find its eigenvalues. What would be the generalization to an n × n matrix? 4C-4. Show that if �� is an eigenvector of the matrix A, associated with the eigenvalue m, 2then �� is also an eigenvector of the matrix A2 , associated this time with the eigenvalue m . (Hint: use the eigenvector equation in 4F-3.) 4C-5. Solve the radioactive decay problem (4B-7) using eigenvectors and eigenvalues. 4C-6. Farmer Smith has a rabbit colony in his pasture, and so does Farmer Jones. Each year a certain fraction a of Smith’s rabbits move to Jones’ pasture because the grass is greener there, and a fraction b of Jones’ rabbits move to Smith’s pasture (for the same reason). Assume (foolishly, but conveniently) that the growth rate of rabbits is 1 rabbit (per rabbit/per year). a) Write a system of ODE’s for determining how S and J, the respective rabbit popu-lations, vary with time t (years). 1b) Assume a = b = 2 . If initially Smith has 20 rabbits and Jones 10 rabbits, how do the two populations subsequently vary with time? c) Show that S and J never oscillate, regardless of a, b and the initial conditions. B B1 2 x1)4x1 ( x 1 4C-7. The figure shows a simple feedback loop. Black box B1 inputs x1(t) and outsputs 1 4 (x1 � − x1). xx x22Black box B2 inputs x2(t) and outputs x2 � − x2. 2 If they are hooked up in …
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