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18.034 Honors Differential Equations Spring 2009
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18.034 Problem Set #3
1. This problem pertains to the differential equation y + ω2y = sin ω0t, where ω = 0 and ω0 is close to but different from
sin 0
(a) Verify that is a particular solution.

(b) As ω0 → ω show that one of the initial conditions becomes infinite.
sin ω0t − sin ωt
(c) Check that y2(t) = ω2 − ω2 is the particular solution for which the initial conditions remain finite as
(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.
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