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(MATH150)[2011](s)final~2083^_64129.pdf
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Final Exam MATH 150: Introduction to Ordinary Differential Equations 25th May 2011
Answer ALL questions using the required method if specified Full mark: 100 Time allowed C 2.5 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 (15)
2 (15)
3 (15)
4 (20)
5 (15)
6 (20)
Total

For the equation: dy ty (4 .y)
=
dt 1+t
(1)
Find out the general solution = ()y >0

yytin its explicit form for and t >0.

(2)
Find out the particular solution of the above equation with the initial condition y(0) =2.

(3)
From the general solution in (1), determine the limiting value of y as t +.


Find the solution x= xt() of the initial value problem:
.2t 2t
x. 4x + 4x= e + 2te , x(0) =1 and x(0) = 4. (You must use the method of undetermined coefficients.)
= yyy () = 0.
Find the solution yy()tof the initial value problem:
+ 2 + 2 = cos t+ t. 2, y(0) = 0 and y(0)
For the initial value problem: yyg (),
+= t
.3, 0 <

t
gt(). () t, t , y(0) 0 and y(0) 1

with =.cos 2 < 2 = =. .
.0, 2<
t
(1)
Graph the function g()t. (A sketch is enough.)

(2)
Express the function g(t)in terms of step (Heaviside) functions.

(3)
Take the Laplace Transform of g(t).

(4)
Find the solution yy()tof the above initial value problem.


=
Find the solution of the initial value problem:
.dx
=2x .5y
.
dt .x (0). .0.
. , with . .=...
dy 4 y 0
.= x .2y .(). .1.
.
.dt 5
(Attention: Solve the problem using matrices) For the system of first order linear equations

=X
XA
. 11 . .x (t ). dX(t ). 2 . 8
with X()=, Xt = , and A=.. .yt(). dt ..1 .
t .. () ...
2
..
. 2. (Attention: Answer the following questions using matrices)
(1)
Find out the real-valued general solution of the above system.

(2)
Determine the arbitrary coefficients in the general solution in (1) by .2.


using the initial condition ()=. ..
X0
3
..
(3) Determine the limit of X(t) as t +.