1
MATH 251
Work sheet / Things to know
Chapter 3
1. Second order linear differential equation
Standard form:
What makes it homogeneous?
We will, for the most part, work with equations with constant coefficients only.
What is the standard form of a second order linear equation with constant coefficients?
Ex
.
Can you think of any function(s) that satisfy each equation (w/ constant coefficients)
below?
(a)
y
″  25
y
= 0
(b)
y
″ + 25
y
= 0
(c)
y
″  25
y
′ = 0
The example (c) above is an instance of a second order linear equation with the
y
±term
missing.
It is essentially a first order linear equation in disguise.
All equations of this
type can be solved by changing it into a first order equation with the substitutions
u
=
y
′
and
u
′ =
y
″, then use the integrating factor method to solve for
u
, and integrate the result
to find
y
.
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2.
The characteristic equation
Given the equation
ay
″ +
by
′ +
cy
= 0, what is its characteristic equation?
Any root,
r
, of the characteristic equation has the property that
y
=
e
rt
always satisfies the
equation above.
Therefore,
y
=
e
rt
will be a particular solution for each root
r
.
Consequently, an important formula to remember for this class is (surprisingly) the
quadratic formula:
a
ac
b
b
r
2
4
2

±

=
Note that the characteristic equation method does
not
require the given differential
equation to be put into its standard form first – the quadratic formula simply doesn’t care
whether or not the leading coefficient is 1.
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 Spring '08
 CHEZHONGYUAN
 Math, Differential Equations, Equations, Partial Differential Equations, Quadratic equation, Elementary algebra, characteristic equation, order linear equation

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