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 9th, 2009 1. Consider the differential equation y�� + y = h(t) − h(t − c) for c > 0. (a) Use the Laplace transform to find the rest solution. (b) Show that y and y� are continuous at t = c but y�� is not. 2. Consider the equation y(n) = δ, where y(t) = 0 for t < 0. Suppose that y is “maximally regular” at 0, i.e., as many derivatives of y as possible are continuous at 0. Show that y(n−1) has a jump of magnitude 1 at t = 0. 3. (a) For n ≥ 0, what is the action of the distribution δ(n) on a test function φ? (b) Explore the continuity of rest solutions to y���(t) = f(t) for the choices t, 1, δ(t), δ�(t), δ�� of f (t). 4. The following boundary-value problem models the equation of the central line y(x), 0 ≤ x ≤ 2, of a uniform weightless beam anchored at one end and carrying a concentrated load at its center. y(iv ) = 6δ(x − 1), y(0) = y�(0) = y��(2) = y���(2). (a) With y��(0) = 2a, y���(0) = 6b, find y via Y (s). (b) Determine a and b from the boundary conditions at x = 2.
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