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 #3 Due by Friday, March 6, 2009, by NOON 1. This problem pertains to the differential equation y�� + ω2y = sin ω0t, where ω =� 0 and ω0 is close to but different from ω. sin ω0t(a) Verify that y1(t) = is a particular solution. ω2 − ω2 0 (b) As ω0 → ω show that one of the initial conditions y1(0) or y�(0) becomes infinite. 1(c) Check that y2(t) = sin ω0t − sin ωt is the particular solution for which the initial conditions ω2 − ω02 remain finite as ω0 → ω. (d) By l’Hospital’s rule show that the limit as ω0 ω of y2(t) gives a particular solution of y�� + ω2y = sin ωt. → 2. Let f(x) and g(x) be two solutions of the differential equation y� = F (x, y) in a domain where F satisfies the condition∗: y1 < y2 implies F (x, y2) − F (x, y1) ≤ L(y2 − y1). Show that |f(x) − g(x)| ≤ e L(x−a)|f(a) − g(a)| if x > a. 3. Very that (sin x)/x, x satisfy the following equations, respectively, and thus obtain the second solution. (a) xy�� + 2y� + xy = 0 (x > 0), (b) (2x − 1)y�� − 4xy� + 4y = 0 (2x > 1). 4. (a) Birkhoff-Rota, pp. 57, #4. (Typo. I(x) = q − p2/4 − p�/2.) (b) Birkhoff-Rota, pp. 57, #7(a). (Use part (a) instead of #6 as is suggested in the text.) (c) Birkhoff-Rota, pp. 57, #7(b). 5. Let (cosh x)y�� + (cos x)y� = (1 + x2)y for a < x < b and let y(a) = y(b) = 1. Show that 0 < y(x) < 1 for a < x < b. 6. (a) Birkhoff-Rota, pp. 75, #3, (b) Birkhoff-Rota, pp. 75, #4. ∗It is called a one-sided Lipschitz condition. 1
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