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
will reduce this equation to z00 + B(x)z = 0, where B(x)
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 will describe a general method, which
will rely on redu
Section 18
Special types of non-homogeneous linear
equations
We will consider non-homogeneous higher order linear equations with constant coefficients,
y(n) + an1 y(n1) + . . . + a1 y + a0 y = q(x),
in the case when q(x) is a linear combination of functio
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 the general solution of this equation and we
already kn
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 the operation of taking a derivative as D =
dk
Dk = dx
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 some ad hoc way. In some cases, one can also appeal to T
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), . . . , yn (x)T , Q(x) = (Q1 (x), . . . , Qn (x)T ,
a
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 the kth derivative of y(x). If q(x) 0 then we call it a
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: when the lines a1 x + b1 y + c1 = 0 and a2 x + b2 y + c2
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 two complex roots z = i0 , the general solution of this
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 4y2 + 8y3 = 0
Dy2 4y3 = 0
y2 + (D 4)y3 = 0
using the di
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 matrix
exponential.
Matrix exponential. Let us define exp
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 +
y = 0,
x
x2
2
2
the coefficients 1x and x xn
are not
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 this equation as
1
x2 n2
y + y +
y = 0,
x
x2
2
2
the c
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 identically equal to zero.
c d
Degenerate case: det(A) =
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,
there exists a unique solution y = y(x) for any initial
Section 27
Predator-prey Lotka-Volterra 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 numbers. Their population dynamics will be
modelled by t
Section 26
Competitive Lotka-Volterra 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 the following Lotka-Volterra equations,
dx
= x(a0 a1 x a
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 at the origin under small changes of initial conditions.
Section 19
Special methods for non-homogeneous,
non-constant coefficients linear equations
In this section we will give two special, non-general, methods that will work assuming that we
already have some partial information.
Method of variation of paramet
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 is unique, at least in a small neighbourhood
of (x0 , y0
Section 11
Picards Method of Successive
Approximations
Suppose that f (x, y) is continuous on the domain D and satisfies y-Lipschitz condition
| f (x, y1 ) f (x, y2 )| K|y1 y2 |
as in the example in the last section. We already know in this case that a so
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 to be vectors in Rn . In other words, y = (y1 , . . . ,
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)
?=i=(+,1;,i)
r
re
call
?, =
al
+
l-
tz
v'l= +'+, . +-+:
OireJio'na1 3"ir".*iues
oofG)
AD
fl':t
=
l;-
tr,s-zO
$
+lne
vg(& ,*/)
G*ditnt
a[ = li- +G+ tas)-+(4 = t.vcfw_
As
AsrO
AS
r fax+ ?fav = '-f u'os +
a:/-l
axolt
li- af = r,3f + u, cfw_ = t.vf
since Af
a5-, o
slopes
AS
in
4u'o'
al
A1<
q.l) direc*io.as c,"e kr
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 of the following equation:
y y 2y = tet
(3) Mark true or
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 false. If true, give an argument why, if false,
give a
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 z
F1 F2 F3
+
+
,
x
y
z
F = (F1 , F2 , F3 )
F3 F2 F1 F3