18.03SC Unit 1 Exam Solutions
.
1. (a) x (t) = number of rats at time t; t measured in years. x = k x. So x (t) = x (0)ekt .
e x (0) = x (1) = x (0)ek implies k = 1.
.
x
x
x = 1
x.
(b) x = k 1
R
1000
.
.
x
(c) x = 1
x a. The pest control people hope for
18.03SC Unit 1 Practice Exam and Solutions
1. A certain computer chip sheds heat at a rate proportional to the difference between its
temperature and that of its environment.
(a) Write down a differential equation controlling the temperature of the chip,
18.03SC Final Exam Solutions
1. (a) The isocline for slope 0 is the pair of straight lines y = x. The direction eld along
these lines is at.
The isocline for slope 2 is the hyperbola on the left and right of the straight lines.
The direction eld along thi
18.03SC Unit 3 Exam Solutions
1. (a) The miminal period is 2.
(b) f (t) is even.
(c) x p (t) =
1
cos( t)
cos(2 t)
cos(3 t)
+
+
+
+
2
2 2 )
2 4 2 )
2
n 2( n
4( n
8( n 9 2 )
(d) There is no periodic solution when n = 0, , 2 , 3 , . . .
or
2. (a)
or
(b)
1
1
18.03SC Unit 2 Exam Solutions
2
1. (a) The characteristic polynomial is p(s) = s2 + s + k = s + 1 + k 1 . This has a
2
4
repeated root when k = 1 .
4
(b) If k is larger, the contents of the square root become negative and the roots become
non-real: so un
18.03SC Unit 2 Practice Exam and Solutions
Study guide
.
1. Models. A linear differential equation is one of the form an (t) x (n) + + a1 (t) x +
a0 (t) x = q(t). The ak (t) are coefcients. The left side models a system, q(t) arises
from an input signal,
Variables and Parameters
1.
Independent and Dependent Variables
When we write a function such as
f ( x ) = 3x2 + 2x + 1
we say that x is an independent variable: it can be freely set to any value
(or any value within the given domain) and the value of the
y
Practice Final Exam
1
m=1 m=0
m=1
dy
y
= + 3 x:
dx
x
a) Sketch the direction eld for this DE, using (light or dotted)
isoclines for the slopes -1 and 0.
1. For the DE
See picture at right
y
Isoclines: y = x + 3x = m y = 3x2 m x.
(Note problem at (0, 0).
Differential Equations
1. Denition of Differential Equations
A differential equation is an equation expressing a relation between a
function and its derivatives. For example, we might know that x is a func
tion of t and
.
.
x + 8x + 7x = 0.
(1)
or perhaps
Notations for Derivatives
We will write
dy
, y and D y
dx
to all mean the derivative of y with respect to x. Only the rst one species
the independent variable x. In the other two you can only determine the
independent variable from context.
When the indep
The Exponential Function
Of primary importance in this course is the exponential function
x (t) = e at ,
where a is a constant. We will assume you are completely familiar with the
properties and graphs of this function.
Properties:
1. e0 = 1.
2. e at+c
Solution to an ODE
Quiz: Solution to an ODE.
Which of the following is a solution to the ODE dy/dx = 2y + 1?
Choices:
a) y = ce2x 1.
b) y = x2 + x + c.
c) y = e x/2 + c.
d) y = ce2x 1/2.
e) y = e2x + c.
f) None of the above
Answer: (d)
This is a little lo