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Unformatted text preview: MATHEMATICS 201, SECTIONS [F01] & [F02] Name: Reg. No.: Instructor: Dr. R. Illner [F01], Dr. B. Khouider [F02]
INIPORTANT: Please circle your tutorial section: TFOI TFD2
TO BE ANSWERED ON THE PAPER Duration: 3 hours STUDENTS MUST COUNT THE NUMBER OF PAGES IN THIS EXAMINATION PA—
PER BEFORE BEGINNING TO WRITE, AND REPORT ANY DISCREPANCY IMME—
DIATELY TO THE INVIGILATOR. THIS EXAMINATION PAPER HAS 10 PAGES plus COVER. 0 NO CALCULATORS, COL/lZPUTERS, BOOKS OR NOTES ARE ALLOW‘E
 Show all your work — substantially correct solutions MAY receive partial credit. 0 Correct answers only, without adequate supporting computations will NOT receive full
credit. m— 1. Find both singular solutions of z’zxz—Zr—S. 2. Find the general solution of (Hint: It is a separable equation). 4. A tank contains 200 liters of ﬂuid in which 30 grams of salt is dissolved. Brine con—
taining 1 gram of salt per liter is then pumped into the tank at a rate of 4 l/mjn.; the
well—mixed solution is pumped out at the same rate. Find the amount A (t) of grams
of salt in the tank at all times t. 23" = —a:c, Where a > 0 is a constant. After 2 days there are 1,000 grams, after 7 days there are
300 grams. How many grams were there initially? 6. Find a fundamental set for
cc”—5m'+4a:=0 (Solve problem 6 ﬁrst). 8. Solve the initial value problem y” +4y = *2 909%. CC Ur} = La” 1:"
n=0 is a solution of
:C”+t:c'+:c=0 3:(0)=1, $’(D)=O. Find (10, a1, a2, a3 and a4. i533+t
$+3t mm 10. Suppose that a: = a: (t) is a solution of as" = then solves . Show that u (t) = the separable equation
du 1 1 — U2 E t 3+u‘ 11. Consider the 2 X 2 system of differential equations , _ 2 3 sint _ :31 (t)
X—(—1 «—2)X+<—2cost)’ X_(3:2(t) ' (1)
(3) Find the eigenvalues and the associated eigenvectors of the matrix A = < _? _: (c) A particular solution of the system in (1) is of the form Determine a system of equations for the constants a1, bl, a2 and b2.
DO NOT SOLVE THE SYSTEM y"+4y=f(t), y(0)=0
y’(0)=—1 1, 0£t<1
Ht):
0, 1321 Where ylkbj=£ ana form a fundamental set of solutions for the D.E. tgy”+ty’—y=0, t>0. (b) Find a particular solution for
t2 y”+ty’~*y=4t6, t>0. Warning or Hint: The D.E. is not in normal form. where A is a constant matrix. Suppose that A E R, V % O and U # 0 are such that
AV=AV and (A—AI)U=V (such V and U can exist when A is a multiple root of the characteristic polynomial).
Show that then
X (t) = e)“ U + ta“ V is a solution of X ’ = AX. ...
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 Fall '10
 STEACY
 Math, separable equation, Dr. R. Illner

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