1
=
y
and
x
2
=
y
0
Obtain
dynamical system
˙
x
1
=
x
2
˙
x
2
=
f
(
t, x
1
, x
2
)
The
state variables
are
y
and
y
0
, which have solutions producing
trajectories
or
orbits
in the
phase plane
For movement of a particle, one can think of the DE governing the
dynamics produces by Newton’s Law of motion and the
phase plane
orbits
show the
position
and
velocity
of the particle
Joseph M. Mahaffy,
h
[email protected]
i
Lecture Notes – Second Order Linear Equations
— (6/32)

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Introduction
Theory for
2
nd
Order DEs
Linear Constant Coefficient DEs
Second Order Differential Equation
Dynamical system formulation
Classic Examples
Classic Examples
Spring Problem
with mass
m
position
y
(
t
),
k
spring constant,
γ
viscous damping, and external force
F
(
t
)
Unforced, undamped oscillator,
my
00
+
ky
= 0
Unforced, damped oscillator,
my
00
+
γy
0
+
ky
= 0
Forced, undamped oscillator,
my
00
+
ky
=
F
(
t
)
Forced, undamped oscillator,
my
00
+
γy
0
+
ky
=
F
(
t
)
Pendulum Problem
- mass
m
, drag
c
, length
L
,
γ
=
c
mL
,
ω
2
=
g
L
, angle
θ
(
t
)
Nonlinear,
θ
00
+
γθ
0
+
ω
2
sin(
θ
) = 0
Linearized,
θ
00
+
γθ
0
+
ω
2
θ
= 0
RLC Circuit
Let
R
be the resistance (ohms),
C
be capacitance (farads),
L
be inductance (henries),
e
(
t
) be impressed voltage
Kirchhoff’s Law for
q
(
t
), charge on the capacitor
Lq
00
+
Rq
0
+
q
C
=
e
(
t
)
,
Joseph M. Mahaffy,
h
[email protected]
i
Lecture Notes – Second Order Linear Equations
— (7/32)

Introduction
Theory for
2
nd
Order DEs
Linear Constant Coefficient DEs
Existence and Uniqueness
Linear Operators and Superposition
Wronskian and Fundamental Set of Solutions
Existence and Uniqueness
Theorem (Existence and Uniqueness)
Let
p
(
t
)
,
q
(
t
)
, and
g
(
t
)
be continuous on an open interval
I
, let
t
0
∈
I
,
and let
y
0
and
y
1
be given numbers. Then there exists a unique
solution
y
=
φ
(
t
)
of the
2
nd
order differential equation:
y
00
+
p
(
t
)
y
0
+
q
(
t
)
y
=
g
(
t
)
,
that satisfies the initial conditions
y
(
t
0
) =
y
0
and
y
0
(
t
0
) =
y
1
.
This unique solution exists throughout the interval
I
.
Joseph M. Mahaffy,
h
[email protected]
i
Lecture Notes – Second Order Linear Equations
— (8/32)

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Introduction
Theory for
2
nd
Order DEs
Linear Constant Coefficient DEs
Existence and Uniqueness
Linear Operators and Superposition
Wronskian and Fundamental Set of Solutions
Linear Operator
Theorem (Linear Differential Operator)
Let
L
satisfy
L
[
y
] =
y
00
+
py
0
+
qy
, where
p
and
q
are continuous
functions on an interval
I
. If
y
1
and
y
2
are twice continuously
differentiable functions on
I
and
c
1
and
c
2
are constants, then
L
[
c
1
y
1
+
c
2
y
2
] =
c
1
L
[
y
1
] +
c
2
L
[
y
2
]
.
Proof uses linearity of differentiation.
Theorem (Principle of Superposition)
Let
L
[
y
] =
y
00
+
py
0
+
qy
, where
p
and
q
are continuous functions on
an interval
I
. If
y
1
and
y
2
are two solutions of
L
[
y
] = 0
(
homogeneous equation
), then the linear combination
y
=
c
1
y
1
+
c
2
y
2
is also a solution for any constants
c
1
and
c
2
.

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