Exam 2 November 23, 2015 1. Find any matrix function A(t) such that the exponential function t exp A(x) ds 0 is not a matrix solution of the system x = A(t)x. However, show that the given exponential formula is a solution in the scalar case (A(t) R), and dene when it is a solution in the matrix valued case (A(t) Rnn ). 2. Compute the fundamental matrix x = Ax where 1 A := 0 0 solution at t = 0 for the system 2 3 1 4 0 1 3. Use this fundamental matrix solution to nd the solution of x = Ax + f (t) where f (t) = (0, t, 0)T . 4. Prove that if is a real number and A is an n n real matrix such that Av, v v 2 for all v Rn , then etA et for all t 0. Hint: Consider the dierential equation x = Ax and the inner product x, x . 1