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18.034 Honors Differential Equations
Spring 2009
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18.034 Solutions to Problemset 3
Spring 2009
1. (b) y1� (0) =
(c) y2� (0) =
ω
1
ω+ω0 ω−ω0
− ω+1ω0 →
(d) lim y2 (t) = −
ω0 →ω
→ +∞ as ω0 → ω.
− 21ω .
1
t cos ωt.
2ω
2. Birkhoff-Rota, pp. 28, Theorem 5.
3. (a) c1 cosx x + c2 sinx x
(b) c1 x + c2 e2x
4. (c)
1
n(n + 1)
+
≥ (n + y2 )2 . Compare the solution with
2
2
(1 − x )
1 − x2
cos(n + y2 )x.
5. Suppose u(x) > 1 at some point a < x < b. Then u takes a positive
maximum > 1 at a < c < b. Observe that u(c) > 1, u� (c) = 0 and
u�� (c) ≤ 0.
At c, the differential equation reduces to
(cosh c)u�� (c) = (1 + e2 )u(c)
≤0
>0
But this is a contradiction.
The case u(x) < 0 at some point is completely analogous.
6. (a) c1 ex + c2 e−x + c3 eix + c4 e−ix or c1 ex + c2 e−x + c3 cos x + c4 sin x
√
√
√
√
(1+i)x/ 2 + c e(1−i)x/ 2 + c e(−1+i)x/ 2 + c e(−1−i)x/ 2 or
(b) c1 e√
4
√
√2
√3
√
√
x/
e 2 (c1 cos x/ 2+c2 sin x/ 2)+e−x/ 2 (c3 cos x/ 2+c4 sin x/ 2)
1