Slide 1
Homogeneous Linear Equations
with Constant Coefficients
Differential Equations
Section 4.3
Slide 2
Homogeneous Linear Equations
with Constant Coefficients
A differential equation of the form
can always be solved in terms of elementary
functions of calculus.
If the coefficients are not constant they are
usually much more difficult.
0
=
+
′
+
′
′
cy
y
b
y
a
Slide 3
Example
Can you think of a solution of the differential
equation given below?
How many different solutions can you come up
with just using trial and error?
0
=

′
′
y
y
Possibilities for y:
e
t
, e
t
, ce
t
, ce
t
, A linear
combination thereof
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View Full DocumentSlide 4
General Case
What about the equation
?
We will form the auxiliary equation
am
2
+ bm + c = 0
.
There are three possibilities for the roots
m
1
and
m
2
of this
equation:
m
1
and
m
2
are real and distinct
m
1
and
m
2
are real and equal
m
1
and
m
2
are conjugate complex numbers
0
=
+
′
+
′
′
cy
y
b
y
a
Slide 5
Case I:
Distinct Real Roots
In this case we find two solutions,
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 Fall '10
 PamelaSatterfield
 Differential Equations, Linear Equations, Equations, Derivative, Complex number, real solutions, Homogeneous Linear Equations

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