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(MATH150)midterm2007Fall.pdf
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J. R. Chasnov
24 October 2007
Answer ALL questions Full mark: 50; each question carries 10 marks. Time allowed C one hour
Directions C This is a closed book exam. You may write on the front and back of the exam papers.
Student Name:
Student Number:
x 1 x
To solve y + p(x)y = g(x),y(x0 )= y0 , let = exp(pdx). Then, y =(y0 +gdx).
x0 x0
Question No. (mark) Marks
1 (10)
2 (10)
3 (10)
4 (10)
5 (10)
Total
Find the solution y = y(x)of thefollowinginitial valueproblem:
22
y = xy, y(.1) =2.
Find the solution y = y(x)for x> 0 of
3
xy . 2y = x cosx,
that passes throught the point(x,y)=(/2,0).
You borrow S0 = $100 000 from the bank at r = 12% interest for T = 3 years. Assume continuous
compounding and repayment of the loan.
(a)(4pts) Determinethedi.erentialequationforthe amountleftontheloan S = S(t) at time t given a
constant repayment rate of k and an interest rate r.
(b)(4pts) Assuming theloanispaidback aftertime T, solve the di.erential equation to determine k in
terms of S0 , r and T.
(c)(2pts)Forthespeci.cloanabove, whatisthetotal amountof moneypaidbacktothebank?
Find the solution x = x(t)of
.t
x+ x =2e, x(0)=0,xB(0)=0.
Find a particular solution of
x+2Bx +x =2e .t .