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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 Problem Set #4 Due by Friday, March 13, 2009, by NOON 1. (a) Show that if u1 is a solution of the third-order variable-coefficient linear differential equation u��� + p1(x)u�� + p2(x)u� + p3(x)u = 0, then the substitution u(x) = u1(x)v(x) leads to the second-order differential equation for v�: u1v��� + (3u�1 + p1u1)v�� + (3u��1 + 2p1u�1 + p2u1)v� = 0. (b) Verify that u1(x) = ex is a solution of the differential equation (2−x)u���+(2x−3)u��−xu�+u = 0. Use the method in part (a) to find the general solution of the differential equation. 2. Find a particular solution of the differential equation x 2 u�� + (1 − α − β)xu� + αβu = x 2f(x) (a) when α =� β and (b) when α = β. 3. Birkhoff-Rota, pp. 62, #2. 4. (a) Find annihilators of xmeαx , xm sin βx, and eαx cos βx. (b) Find the general solution of (D2 + 1)(D − 3)2u = 12e3x(10x + 1). 5. Consider the nth order linear homogeneous differential equation u(n) + a1u(n−1) + + an−1u� + anu = 0 · · · with real constant coefficients. (a) Show that if the differential equation is asymptotically stable then a1, a2, , an > 0.· · · (b) Show that the converse to part (a) is true if all roots of its associated characteristic polynomial are real. (Hint. Assume it is false, and seek for a contradiction.) (c) Show by a counter example that the converse to part (a) is in general false. 6. (a) Show that the function y2 is not Lipschitzian on −∞ < y < ∞. Discuss how this failure of the Lipschitz condition is reflected in the behavior of the initial value problem y� = y 2 , y(0) = y0 > 0. (b) Show that the function y2/3 is not Lipschitzian in any strip |y| < h containing the origin. Discuss how this failure of the Lipschitz condition is reflected in the behavior of the initial value problem y� = y 2/3 , y(0) = 0. 1


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MIT 18 034 - Problem Set #4

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