MIT18_03S10_4ex

MIT18_03S10_4ex - 4 Linear Systems 4A Review of Matrices...

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�� 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 1 A = 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 A 3 (= A · A · A ). 1 1 4A-6. For what value of c will the vectors x 1 = (1 , 2 , c ) , x 2 = ( 1 , 0 , 1) , and x 3 = (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 c 1 x 1 + c 2 x 2 + c 3 x 3 = 0 4B. General Systems; Elimination; Using Matrices 4B-1. Write the following equations as equivalent first-order systems: d 2 x dx y �� x 2 a) + 5 + tx 2 = 0 b) y + (1 x 2 ) y = sin x dt 2 dt 4B-2. Write the IVP y (3) + p ( t ) y �� + q ( t ) y + r ( t ) y = 0 , y (0) = y 0 , y (0) = y 0 , y �� (0) = y 0 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 = 4 x y, y = 2 x + y , 2 t a) using matrix notation, verify that x = e 3 t , y = e 3 t and x = e , y = 2 e 2 t are solutions; b) verify that they form a fundamental set of solutions i.e., that they are linearly independent; 1
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2 18.03 EXERCISES c) write the general solution to the system in terms of two arbitrary constants c 1 and c 2 ; write it both in vector form, and in the form x = . . . , y = . . . . 1 3 4B-5. For the system x = A x , where A = 3 1 , 1 2 t a) show that x 1 = 1 e 4 t and x 2 = e form a fundamental set of solutions 1 1 (i.e., they are linearly independent and solutions); 5 b) solve the IVP: x = A x , 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 = A x , where A = , x = , a, b constants .
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