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 14th, 2009 1. Consider the problem of recovering the differential operator T = p(D) from knowledge of the solution y(t) to T y = 0 with initial conditions y(0) = 1, y�(0) = y��(0) = = y(n−1)(0) = 0.· · · (Since p(D) can at most be determined up to a multiplicative constant, we assume p(D) to be monic.) (a) Check that L[y(j)] = sj L[y] − s(j−1) for j = 1, 2, . . . , n, and conclude that Y (s) gives no information about P (s) if P (0) = 0 (except that, in fact, P (0) = 0). (b) Show that if P (0) = 0, however, � P (s) can be fully determined from Y (s). 2. Suppose A : (a, b) → Mn is an n × n matrix and det A(t) = 0 for all � t ∈ I. Compute B�(t) where B(t) = A−1(t). 3. Take the second order equation x�� + p(t)x� + q (t)x = 0 and change it to a first-order system v� = Av. in the usual way. Show that the Wron-skian of two solutions x1, x2 to the original equation is the same as the Wronskian off the two corresponding solutions x1, x2 of the system. Show that Abel’s formula for the Wronskian of two (scalar) solutions to the sec-ond order ODE agrees with what one obtains for the Wronskian of two linearly independent solutions in the vector formulation when one applies the identity (det Y (t))� = det Y (t) tr Y � for an smooth invertible family of matrices Y (t). 14. For the system y1�= 3y1 + 2y2, y2�= −2y1 − y2, find the unique fundamental matrix U (t) satisfying U(0) = I.
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