MIT OpenCourseWarehttp://ocw.mit.edu 18.034 Honors Differential Equations Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.18.034 Recitation: April 16th, 2009 1. Suppose A is an n × n matrix and y1(t), y2(t), . . . yn(t) are solutions to y� = Ay. Show that the set if {yi(t0)}in =1 is linearly independent at some time t0, then to any other solution y(t) there correspond constants ci so that y(t) = c1y1(t) + c2y2(t) + . . . + cnyn(t) (i.e., the set {yi(t)}n i=1 constitutes a basis of solutions). 2. Let A be an n × n matrix. (a) Suppose v1 and v2 are eigenvectors of A corresponding to the eigen-values λ1 and λ2, respectively. If λ1 �= λ2, show that v1 and v2 are linearly independent. (b) Assume now that n = 2. If pA(λ) = (λ − λ1)2, show that either A = λ1I, or there is a unique eigenvector v1 associated to λ1 and a vector v2 satisfying (A − λ1)v2 = v1. (c) For A as in the latter alternative in (2), show that the general solution to d y = Aydt is given by y = eλ1t(c1t + c2)v1 + c1eλ1tv2. 3. For the system y1�= 3y1 + 2y2, y2�= −2y1 − y2, find the unique fundamental matrix U(t) satisfying U(0) = I. 4. Under what conditions on the trace and determinant of the 2 × 2 matrix A will all solutions to the equation y� = Ay satisfy limt→∞ |y(t)| = 0?
View Full Document
Unlocking...