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Unformatted text preview: Tuesday, 31 May 2005
Math 5A, Final Exam Review 105
1. Let A = 2 1 6 .
a. Compute det A.
b. Compute A−1 . Verify that this is, indeed, the inverse of A. 5 6 .
c. Solve Ax = b, where b =
2. Suppose for a mass-spring system, one obtains the diﬀerential equation
x + 4x + 5x = 0, with initial values x(0) = 1 and x (0) = 0.
a. Solve the equation. Write it in its phase-amplitude form.
b. What is the earliest time at which this system crosses its equilibrium position?
3. a. Solve y − 4y = 6e2t .
b. Write the aforementioned diﬀerential equation as an equivalent system of ﬁrst
order diﬀerential equations. 4. Given: dx
dt = 5x + 4y, x(0) = 1
= −4x + 5y, y (0) = −3 a. Solve it.
b. What is the behavior of its equilibrium point (0,0)?
5. Given: dx
dt = 3 sin x + y 3
=y a. Find its points of equilibrium.
b. Linearise the system at (0,0). Determine its stablility.
6. True or False?
a. AB = BA for any matrices A, B.
b. det AB = (det A)(det B)
c. If A2 = 0, then A = 0.
d. Let c be a scalar. If λ is an eigenvalue for A, then cλ is an eigenvalue for cA. ...
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- Fall '07