Section 4.3
Linear Homogeneous Equations
with Constant Coefficients
In this section we look for solutions to linear, homogeneous equations with constant coefficients
where a, b, and c are constants.
ay + by + cy = 0
To solve the 2nd order equation with co

Section 4.2
Theory of Second Order Linear
Homogeneous Equations
In this section we consider
y + p (t ) y + q (t ) y = g (t )
with initial conditions
y (t0 ) = y0 ,
y (t0 ) = y1
where p, q, and g are continuous on an open interval I.
This is a second order

Chapter 4
Second Order Linear
Equations
Section 4.1
Definitions and Examples
A second order ordinary differential equation has the general form
y = f (t , y , y )
where f is some given function.
This equation is said to be linear if f is linear in y and y

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3. Determine the largest interval in which the IV? is certain to have a unique solution.
(x - 2)y" + y + (x - ZXtan x)y = 0, y(3) =1, y(3) = 2 t7; J
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MthSc 208
Review Test 3
March 2013
1. Consider the DE
a. Express the differential equation as a system of DE of order 1 in matrix form.
b. Find the solution to the corresponding homogeneous system of differential equations.
2. Find the solution to the dif

Method of Undetermined Coefficients
Method of Variation of Parameters
1
2
If y and y are fundamental solutions to
where
y + p(t ) y + q (t ) y = g (t )
y (t ) g (t )
y (t ) g (t )
u1 (t ) = 2
dt , u2 (t ) = 1
dt
W ( y1 , y2 ) (t )
W ( y1 , y2 ) (t )
then

Section 3.5
Repeated Eigenvalues
This section deals with two-dimensional linear homogeneous systems with
constant coefficients when the system has repeated roots.
x = Ax
There are two cases to consider:
i. There are repeated eigenvalues with two independe

Section 3.5
Repeated Eigenvalues
This section deals with two-dimensional linear homogeneous systems with
constant coefficients when the system has repeated roots.
x = Ax
There are two cases to consider:
i. There are repeated eigenvalues with two independe

Section 3.4
Complex Eigenvalues
This section deals with systems of differential equations with complex eigenvalues.
Recall that eigenvalues are found by solving the equation
det( A I) = 0
(1)
If the roots of this polynomial are complex they can be express

Chapter 3
Systems of Two First Order Equations
Section 3.1
Systems of Two Linear Algebraic Equations
To solve a system of two linear differential equations with constant coefficients we
use the concepts of linear algebra. In this section we are learning h

Section 4.5
Nonhomogeneous Equations;
Method of Undetermined Coefficients
A nonhomogeneous equation is of the form
y + p(t ) y + q(t ) y = g (t )
where p, q, g are continuous functions on an open interval I.
The equation y + p (t ) y + q (t ) y = 0 is cal

Section 4.6: Forced Vibrations, Frequency
Response, and Resonance
Wecontinuethediscussionofspringmasssystems,andnowconsider
thepresenceofaperiodicexternalforce:
m y (t ) + y (t ) + k y (t ) = F0 cos t
ForcedVibrationswithDamping
Considertheequationbelowfo

Section 4.7
Variation of Parameters
In this section we study a second method to find the particular solution to a
nonhomogeneous second order linear equation. This method is called
Variation of Parameters. The main advantage of this method is that its a g

Section 5.7
Impulse Functions
In some applications, it is necessary to deal with phenomena of an impulsive nature. For example,
an electrical circuit or mechanical system subject to a sudden voltage or force g(t) of large
magnitude that acts over a short

Section 5.6
Differential Equations with
Discontinuous Forcing Functions
In this section we look at some examples of nonhomogeneous differential equations
where the nonhomogeneous term is a discontinuous function.
Example 1: Find the solution to the initia

Section 5.5
Discontinuous Functions and Periodic Functions
In this section we consider one of the most useful applications of the Laplace transform:
solving linear differential equations with discontinuous, periodic, or impulsive forcing
functions. These

Section 5.4
Solving Differential Equations with
Laplace Transforms
To solve a differential equation using Laplace Transforms we follow three steps:
i. Use the linearity of L to transform the IVP in (y (t) in the t-domain to (Y (s) ) the s-domain
ii.
Use a

Section 5.3
The Inverse of the Laplace Transform
The main difficulty in using the Laplace transform method is determining the function
y = (t) such that Lcfw_(t) = Y(s). This is an inverse problem, in which we try to find
such that (t) = L-1cfw_Y(s).
The

Section 5.2
Properties of the Laplace Transform
In this section we will learning some properties of the Laplace Transform that
will help us find the transformation of a differential equation without using the
definition of the Laplace Transform.
1. Shifti

Section 5.5
Discontinuous Functions and Periodic Functions
In this section we consider one of the most useful applications of the Laplace transform:
solving linear differential equations with discontinuous, periodic, or impulsive forcing
functions. These

Test 3
Section 4.1 Definition and Examples:
A second order ordinary differential equation has the general form
y = f (t, y, y ) where f is some given function.
The equation is said to be linear if f is linear in y and y , y = g(t) p(t) y q(t)y. Otherwise

Section 4.4
Mechanical & Electrical Vibrations
Two important areas of application for second order linear equations with constant
coefficients are in modeling mechanical and electrical oscillations. We will study the
motion of a mass on a spring in detail

Section 3.3
Homogeneous Linear Systems with
Constant Coefficients
In this section we study a 2 X 2 system of first order differential equations with
constant coefficients, which can be written as
x1 = a11 x1 + a12 x2
x2 = a21 x1 + a22 x2
Or in matrix form

Section 3.2
Systems of Two First Order Linear
Differential Equations
This section deals with systems of two first order linear differential equations.
This system is expressed as
dx / dt p11 (t ) x + p12 (t ) y + g1 (t )
=
dy / dt p21 (t ) x + p22 (t )

det( A I ) = 0
( A 1I ) v = 0
( A 1I )w = v
x1 (t ) = e 1t v
x 2 (t ) = te 1t v + e 1t w
1 = + i ,
2 = 1 = i
e( +i ) t = e t ei t = e t (cos( t ) + i sin( t )
Solution Behavior of Second Order linear differential equations or systems:
Saddle point
Nodal s

Eulers and Improved Eulers Methods
Method for Linear Equations
Exact equation
Convolution
Solutions to differential equations with complex eigenvalues
Systems of Equations
det( A I ) = 0
Complex Solutions
Given complex eigenvalues
then
Method of Variation