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Unformatted text preview: MATH 2243 FINAL EXAM
FALL 2005 (INSTRUCTORS: S. Joo / M. KURZKE) Name: Discussion Session: (021: Javier Zuniga TTh 10:10, 022: Chung—I H0 TTh 10:10, 023: Javier Zuniga TTh 11:15,
024: Chung~I Ho TTh 11:15 031: Doyoon Kim TTh 2:30,
032: Javier Zuniga TTh 2:30, 033: Doyoon Kim TTh 3:35) ll 4 nun Total
Ill  Exam Rules: 0 Oneline scientiﬁc calculators are allowed. 0 No books or notes. 0 All questions are partialcredit. 0 Put ﬁnal answers in boxes where appropriate (no boxes for 5 and 9). 0 Good Luck! (15 pts) 1. Find the general solution for the differential equation y'+3y=5. (30 pts) 2. Convert the equation
y”yy’+2y'—y2+5y=0 to a corresponding system, and determine the location, type and stability of all equi—
librium points via the linearization of that system at its equilibrium points. system : equilibrium points, types and stability: (25 pts) 3. Determine the kernel, the nullity and the rank of the transformation T : R3 —+ R2 given by
2 2 2
T(v) — (2 3 2) v. (30 pts) 4. Find the general solution of (25 pts) 5. Determine Whether or not the set S = {3t + 15, 3t — 15, tet} is linearly independent on (—00, 00). Is It in the span of S? Justify your answer. (25 pts) 6. Find the general solution of y” + 43/ — 9y 2 0. 25 ts 7. A tank initially contains 10 gal of pure water. A salt solution With concentration
p
5 lb/gal is pumped into the tank at a rate of 2 gal/min, and the well—stirred mixture
ﬂows out at a rate of 4 gal/min. What is the actual amount of salt y(t) in the tank at time t? (20 pts) 8. Solve the initial value problem 1/ 2 63“”, y(0) = 0. (25 pts) 9. Is the set {<:>eRzzw=o} a subspace of R2? Justify your answer. (20 pts) 10. Solve the initial value problem ty' — y = t2 cost, y(7r) :2 3. 10 (30 pts) 11. Find the general solution of the system 11 (25 pts) 12. Compute the inverse of the matrix 111
A=021
101 12 ...
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 Fall '08
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