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(math150)[2011](s)midterm~sfling^_81129.pdf
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Midterm Exam MATH 150: Introduction to Ordinary Differential Equations 9th April 2011
Answer ALL questions Full mark: 100; each question carries 20 marks Time allowed C 1 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:
Question No. (mark) Marks
1 (20)
2 (20)
3 (20)
4 (20)
5 (20)
Total
Find the solutions x= xt() of the initial value problem:
22
1+ xdt. xt1+ t dx= 0 and x(1) = 0, with x 0 and t 1.
() ()
=
Find the solution yy()xof the initial value problem:
y+ yx +1 = 0 and y(0) = 1, with x>. 1.
2 .( )3
x+1
A cup of hot chocolate is initially at 80 C. Being left in a room with a
fixed ambient temperature T = 20 C , the temperature of the hot chocolate
a
changes to 60 C in 5 minutes. Suppose that the cooling of the hot chocolate is governed by the law as follows:
The rate of change of the temperature is proportional to the difference between the current temperature and the (fixed) ambient temperature. And the proportionality coefficient is a constant.
By finding and solving a differential equation for the temperature of the hot chocolate, show how long it takes for the hot chocolate to reach a
temperature at 40 C?
Find the solution x=xt() of the following initial value problem: 12 .+5x=0 with x(0) =.1 and x(0) =1.
xx
Find the solution x= xt() of the following initial value problem:
.2t 3t
x. 4x + 3x=15 e +10e with x(0) =1 and x(0) = 5.