Section 31
Sturm comparison theorem
If we consider the second order linear equation
y00 + a(x)y0 + b(x)y = 0,
then it is easy to check that the change of variables
y(x) = z(x) exp
1
2
Z x
x0
a(t) dt
w
Section 21
Systems of linear equations with constant
coefficients
Let us consider a homogeneous system of linear equations
dy
= Ay,
dx
where y = (y1 , . . . , yn )T and A is a constant n n matrix. We
Section 18
Special types of nonhomogeneous linear
equations
We will consider nonhomogeneous higher order linear equations with constant coefficients,
y(n) + an1 y(n1) + . . . + a1 y + a0 y = q(x),
i
Section 16
Linear homogeneous equations with
constant coefficients
Let us now consider the equation
y(n) + an1 y(n1) + . . . + a1 y + a0 y = 0,
where a1 , . . . an1 are constants. Our goal is to find
Section 17
Differential and polynomial operators
Consider an expression y 3y + 2y. We can write it formally as
! d2
"
!d d
"
d 2y
dy
d
d
3
+
2y
=
3
+
2
y
=
3
+
2
y.
dx2
dx
dx2
dx
dx dx
dx
Let us write
Section 15
Wronskian
In most cases, it will be easier to check linear independence of solutions y1 (x), . . . , yn (x) of the
equation
y(n) + an1 (x)y(n1) + . . . + a1 (x)y + a0 (x)y = 0
by hand, in s
Section 13
Systems of Linear Equations: General
Theory
In this section, we will consider a special linear system of equations given by
dy
= A(x)y + Q(x)
dx
where y and Q are n 1 vectors
y(x) = (y1 (x)
Section 14
Higher Order Linear Equations: General
Theory
In this section, we will study equations of the form:
y(n) + an1 (x)y(n1) + . . . + a1 (x)y + a0 (x)y = q(x),
(14.0.1)
where y(k) (x) denotes t
Section 4
Equations With Linear Coefficients
Here, we will explain how to solve equations of the form
(a1 x + b1 y + c1 ) dx + (a2 x + b2 y + c2 ) dy = 0.
There are two different cases to consider: wh
Section 20
Harmonic motion
Undamped harmonic motion.
We will say that an objects executes a simple harmonic motion if its linear position in time satisfies
d2x
+ 02 x = 0.
dt 2
Since z2 + 02 = 0 has t
Section 23
Solving linear systems by Gaussian
elimination
Example. Let us consider the system
dy1
= 2y1 + 4y2 8y3
dx
dy2
= 4y3
dx
dy3
= y2 + 4y3
dx
that we solved before. Let us rewrite it as
(D 2)y1
Section 22
Matrix exponential
Let us again consider a homogeneous system of linear equations
dy
= Ay,
dx
where y = (y1 , . . . , yn )T and A is a constant n n matrix. We will now solve it using the ma
Section 30
The Bessel equation
We will now find one solution of the Bessel equation
x2 y + xy + (x2 n2 )y = 0
for each integer n 0. As we mentioned before, if we rewrite this equation as
1
x2 n2
y + y
Section 29
The wave equation
As an example of application of series method we will study the Bessel equation
x2 y + xy + (x2 n2 )y = 0,
and we will focus on the case when n 0 is integer. If we rewrite
Section 24
Phase portraits for 2 2 linear systems.
Let us consider a 2 2 system of linear first order equations
dy1
= ay1 + by2
dx
dy2
= cy1 + dy2 .
dx
a b
We will assume that the matrix A =
is not id
Section 28
Series methods
Let us consider the equation
y(n) + fn1 (x)y(n1) + . . . + f0 (x)y = Q(x).
We showed previously that when the functions f0 , . . . , fn1 , Q are continuous on some interval,
Section 27
Predatorprey LotkaVolterra equations.
Let us consider an ecological system of two species, one predator (such as foxes) and the other
one prey (rabbits). Let x and y be their population n
Section 26
Competitive LotkaVolterra equations.
Let us consider an ecological system of two competing species. Let x and y be their population
numbers. Their population dynamics will be modelled by t
Section 25
Critical points and linearization.
Another type of instability.
When we considered various cases in the previous section, we described instability of the long term
behaviour of a solution a
Section 19
Special methods for nonhomogeneous,
nonconstant coefficients linear equations
In this section we will give two special, nongeneral, methods that will work assuming that we
already have s
Section 10
Osgoods Uniqueness Theorem
In the next result, we will describe a condition on f (x, y) under which the solution to a differential
equation y = f (x, y) with initial condition y(x0 ) = y0 i
Section 11
Picards Method of Successive
Approximations
Suppose that f (x, y) is continuous on the domain D and satisfies yLipschitz condition
 f (x, y1 ) f (x, y2 ) Ky1 y2 
as in the example in t
Section 12
Systems of First Order Equations:
Existence and Uniqueness
In this section, we will consider the same differential equation as before,
dy
= f (x, y),
dx
only now we will allow both y and f
ag
Vec*sr Fi"lds
(F,cfw_r,y), f,cryl )
aD
F(ay)
3D
i(*1,") =(r, ,F., Fr)
=
ap;, 4i"lJ
aD Ex
7
=
Cx,Y,')
a
V'r a
.,
v's =Q
rO El, i'= (*,r)
$
f; all cfw_.tnchons ocfw_
(l,x)
J
9'i = s
7 = Q,7,2)
MAT 267F
Practice Term Test
(1) Solve the following IVP:
y = 2 cos2 y sin(2x)
y(0) = 0
What is the interval of existence of the solution?
(2) Using the variation of parameter nd the general solution o
1. (6 pts) Give the denitions of the following notions:
a) Asymptotically stable equilibrium of a system of dierential equations;
b) A homogeneous linear dierential equation;
2. (10 pts) Mark true or
Test 3 Topics
MAT 267 Honors Calculus for Engineers III
HWs 810 (omit only sections 13.6 & 13.7) plus Lectures Notes plus
Memorize!
Gradient, Divergence, Curl
=
f =
F =
F =
,
,
x y z
f f f
,
,
x y
t1
Do,rbk, f'n*eqrcls
E* F;"J
t?,fl
V rrnder
o sy
o <x <l
=
l+
x'+y' cfw_o,
< &
e.
z
Afi) =
=
J6* *'*y")JY
V= J
= .ft1

[;
A(x)
xo
+. t'l:
a
'
*io\y,?
=
fubir,i!
a
o
y') a/
16
rf J(x,i) is con*r