1. Consider the differential equation
y
00 + a1y
0 + a2y = 0,
where a1, a2 ∈ R.
(a) Suppose the roots of the auxiliary equation p(r) = r
2 + a1r + a2
are both real numbers. What condition on these roots (call them
r1 and r2) guarantees that all solutions y of the equation satisfy
lim
t→+∞
y(t) = 0?
(b) Suppose the roots of the auxiliary equation are NOT real numbers.
What condition on these complex conjugate roots guarantees that
all solutions y of the equation satisfy
lim
t→+∞
y(t) = 0?
(c) Suppose that a1 and a2 are both positive numbers. Is it necessarily
true that all solutions y satisfy
lim
t→+∞
y(t) = 0?
Why or why not?
2. Euler’s formula tells us that
e
it = cost + isin t.
(1) Use it to show that y = e
7x
cos(3x) is a linear combination of
the functions f(x) = e
(7+3i)x and g(x) = e
(7−3i)x
.
(2) Use it to show that y = e
7x
sin(3x) is a linear combination of
the functions f(x) = e
(7+3i)
5 years ago
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