Group: Worksheet 12 pt.2 (xx7.3-4) 1. Either solve the following systems of equations or show that there is no solution. (a) 2x1 x2 + 3x3 = 1 x1 + x3 = 2 x1 + x2 + 2x3 = 1 (b) x1 + 2x2 + 2x3 = 2 x1 + 2x2 = 1 x1 + x3 = 1 2. Find the eigenvalues and eigenvectors for the following matrices. (a) (#18 pg 384) 1 i i 1 ! (b) (#18 pg 384) 0 BB @ 3 2 2 1 4 1 2 4 1 1 CC A 3. (#6 pg 389) Consider the vectors x(1)(t) = t 1 ! and x(2)(t) = t2 2t ! (a) Compute the Wronskian of x(1) and x(2). (b) In what intervals are x(1) and x(2) linearly independent? (c) What conclusion can be drawn about the coe cients in the system of homo- geneous di erential equations satis ed by x(1) and x(2)? (Hint: See Theorem 7.1.2 pg 359.) (d) Find the system of di erential equations and verify the conclusions of part (c). 4. (#4 pg 389) If x1 = y and x2 = y0, then the second order equations y00 + p(t)y0 + q(t)y = 0 corresponds to the system x01 = x2 x02 = q(t)x1 p(t)x2: Show that if x(1) and x(2) are a fundamental set of solutions of this system, and if y(1) and y(2) are a fundamental set of solutions of the second order equation, then W[y(1);y(2)] = cW[x(1);x(2)], where c is a nonzero constant. Hint: y(1)(t) and y(2)(t) must be linear combinations of x11(t) and x12(t).
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