Ch 4.8 (4.9 for 8-th edition): Solving systems of Linear
Des by Elimination
In this section, we will solve simultaneous ordinary dierential
equations with constant coecients using the method called
sy
Ch 4.2: Reduction of order
In this section, we will
1. solve a homogeneous linear second-order dierential equation
by a method called reduction of order.
2. revisit the concept of linear independency/
Ch 4.6: Variation of parameters
In this section, we consider a second-order linear DE with constant
coecients
a2 y + a1 y + a0 y = g (x ).
We will also
1. review the Wronskian, W
2. and nd a general s
Ch 3.1 : Modeling with rst-order linear DEs.
In this section, we will solve some of the linear DEs we came up
with in Ch 1.3.
Bacterial Growth
Recall that if P (t ) denote a certain population at time
Ch 2.3 : Linear Equations
In this section, given a 1st order linear dierential equation, we will
1. dene a standard from of the given ODE
2. dene the integrating factor
3. solve the standard form of t
Ch 1.2: Initial-Value problem (IVP)
In this section, we will look at examples concerning simple DEs
with initial conditions. We will then study some uniqueness
theorem regarding when a solution to DE
Section 2.2: Separable variables
In this section, we will
solve a rst-order dierential equation with separable
variables and come up with a solution (or solutions).
Separable Equation
Denition
A rst-o
Ch 4.3 : Homogeneous Linear Equations with Constant
Coecients
In this sections, we look at the following homogeneous linear
higher-order DEs
an y (n) + an1 y (n1) + + a2 y + a0 y = 0
where ai s are re
Ch 4.1: Preliminary Theory - Higher-order Linear
equations
In this section, we will
1. consider a higher-order linear DEs with initial conditions
2. consider the concept of linear independence/depende
Ch. 5.1.2: Spring/Mass systems: Free damped motion
In this subsection, the motion is damped by damping forces. We
will
1. consider the DE of free damped motion
2. consider the three cases : overdamped
Ch 5.1.3 : Spring/Mass Systems : Driven Motion
In this case, we will
1. consider DE of driven motion with damping
2. look at DE of driven motion without damping (undamped
forced motion)
Driven motion
Ch 7.1 : Denition of Laplace transform
Denition
Let f (t ) be a function dened for t 0. Then the integral
L cfw_f (t ) =
e st f (t )dt
0
is called the Laplace transform of f , provided the integral
co
Ch 5.1 : Linear Models: Initial-value problems
In this section, we will consider a linear dynamical system modeled
by the 2nd-order linear DE with constant coecients with initial
conditions:
dy
d 2y
+
Ch 4.4: Undetermined Coecients
In this section, we will consider a non-homogeneous linear DE of
the form
an y (n) + an1 y (n1) + + a1 y + a0 y = g (x )
where ai s are constants.
The idea is that depen
Ch 1.3 : DIFFERENTIAL EQUATIONS AS
MATHEMATICAL MODELS
Population Dynamics:
Let P (t ) denote the total population of humans at time t, then
one can describe the growth of P as
dP
= kP
dt
where k is c