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(MATH150)final2005_Fall.pdf
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Final Exam MATH 150: Introduction to Ordinary Di.erential Equations
J. R. Chasnov
15 December 2005
Answer ALL questions Full mark: 90; each question carries 10 marks. Time allowed C 2 hours
Directions C This is a closed book exam. You may write on the front and back of the exam papers.
Student Name:
Student Number:
Question No. (mark) Marks
1 (10)
2 (10)
3 (10)
4 (10)
5 (10)
6 (10)
7 (10)
8 (10)
9 (10)

Total
(a)(8pts)Solvetheinitial valueproblemfor x 0:
dy 2cos2x
= ,y(0)= .2.
dx 3+2y
(b)(2pts)Determine wherethesolutionattainsitsminimumvalue. Whatisthevalueofy atitsminimum? (a)(6 pts) With , a and b all positive, .nd the solution x = x(t) for t 0 of the following initial value
problem: xB+x =; x(0)=0.
a +be.t
(b)(2pts)Determinethe value of t at which x(t)is maximum. (c)(2 pts)Determine x as t .
Newtons law of cooling states that the temperature of an object changes at a rate proportional to the di.erence between its temperature and that of its surroundings. Suppose that the temperature of a cup of co.ee obeys Newtons law of cooling. If the co.ee has a temperature of 900C when freshly poured, and in a room of 200C cools to 600C in .ve minutes, determine a formula for the temperature of the co.ee at any timet(in minutes).
(a)(8pts)Given , .ndthesolutionof thefollowing initial valueproblem: x+2Bx +x =0,x(0)=1,xB(0)= . (b)(2pts)Determinethe values of such that x(t)for t> 0 becomes negative.
(a)(8pts)Find the solutionfor t 0 ofthefollowinginitial valueproblem:
x+2Bx +2x =0,x(0)=0,xB(0)= u0 > 0. (b)(2pts)Determinethetimeat which themaximumvalueof x occurs.
Find the solution of the initial value problem
x+x = F(t),x(0)=0,xB(0)=0,
where
F0t, if 0 t F(t)= F0(2 . t), if <t 2 0, if t> 2.
Hint: Treat each time interval separately and match the solution in the di.erent intervals by requiring that x and Bx be continuous functions of t.
(a)(8points)By solving thegivendi.erential equationby means of apower series about x0 = 0, .nd the .rst three nonzero terms in each of two linearly independent solutions:
2
y +k2 xy =0,k a constant.
(b) (2 points) Find the .rst four nonzero terms in a power series solution about x0 = 0 that satisfy the initial conditions y(0)=1, y (0)=1.
Determine the solution y = y(x)of the di.erential equation
2
2xy +3xy . y =0, x> 0
given the conditions limx0 xy(x)=2 and y(1)=1.
Consider the system of equations
xB1 = .x1 . x2,xB2 = .x1 . x2.
(a)(2pts)Determinethe values of such that all solutions for x1 and x2 go to zero as t . (b)(6 pts)Solve the initial valueproblem for =4, x1 (0)=1, x2 (0)= s, s constant. (c)(2pts)Determinethe value of s such that x1 and x2 go to zero as t