Notes on Second Order Linear Differential
Equations
Stony Brook University Mathematics Department
1.
The general second order homogeneous linear differential equation with constant co
efficients looks like
Ay
+
By
+
Cy
=
0,
where
y
is an unknown function of the variable
x
, and
A
,
B
, and
C
are constants. If
A
=
0
this becomes a first order linear equation, which we already know how to solve. So we
will consider the case
A
=
0. We can divide through by
A
and obtain the equivalent
equation
y
+
by
+
cy
=
0
where
b
=
B
/
A
and
c
=
C
/
A
.
“Linear with constant coefficients” means that each term in the equation is a constant
times
y
or a derivative of
y
. “Homogeneous” excludes equations like
y
+
by
+
cy
=
f
(
x
)
which can be solved, in certain important cases, by an extension of the methods we will
study here.
2.
In order to solve this equation, we guess that there is a solution of the form
y
=
e
λ
x
,
where
λ
is an unknown constant. Why? Because it works!
We substitute
y
=
e
λ
x
in our equation. This gives
λ
2
e
λ
x
+
b
λ
e
λ
x
+
ce
λ
x
=
0.
Since
e
λ
x
is never zero, we can divide through and get the equation
λ
2
+
b
λ
+
c
=
0.
Whenever
λ
is a solution of this equation,
y
=
e
λ
x
will automatically be a solution of our
original differential equation, and if
λ
is not a solution, then
y
=
e
λ
x
cannot solve the
differential equation. So the substitution
y
=
e
λ
x
transforms the differential equation into
an algebraic equation!
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2
S
ECOND
O
RDER
L
INEAR
D
IFFERENTIAL
E
QUATIONS
Example 1.
Consider the differential equation
y

y
=
0.
Plugging in
y
=
e
λ
x
give us the associated equation
λ
2

1
=
0,
which factors as
(
λ
+
1
)(
λ

1
) =
0;
this equation has
λ
=
1 and
λ
=

1 as solutions. Both
y
=
e
x
and
y
=
e

x
are solutions
to the differential equation
y

y
=
0. (You should check this for yourself!)
Example 2.
For the differential equation
y
+
y

2
y
=
0,
we look for the roots of the associated algebraic equation
λ
2
+
λ

2
=
0.
Since this factors as
(
λ

1
)(
λ
+
2
) =
0, we get both
y
=
e
x
and
y
=
e

2
x
as solutions to
the differential equation. Again, you should check that these are solutions.
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 Fall '07
 GuanYuShi
 Differential Equations, Calculus, Equations, real solutions

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