MATH 2214 Midterm: MATH 2214 Virginia Tech 4.8 Exam
Document Summary
Solutions 4. 8: a) the eigenvalues are i with eigenvectors (1, i) and (1, i). The com- plementary solutions is c1 (cid:18) cos t. Sin t(cid:19) + c2 (cid:18) sin t cos t(cid:19) . (cid:18) 0. 1: we get which leads to c) Sin t(cid:19) + c2 (cid:18) sin t cos t(cid:19) : the initial conditions yield. 2 + c1 = 1, 1 + c2 = 1, hence c1 = 1, c2 = 2: a) the eigenvalues are 1, with eigenvectors (1, 1) and (1, 1). 0(cid:19) = (cid:18) 0 a = (cid:18) 0 1. 1(cid:19) : the initial conditions lead to c1 + c2 = 2, c1 c2 = 1. This has the solution c1 = 1/2, c2 = 3/2: the fundamental matrix can be read o from problem 2a: Sin t sin t cos t(cid:19) . The variation of paramters formula yields y(t) = (t)(cid:18) 0. = (cid:18) sin t cos t 1.
