Review of Ordinary Differential Equations
Definition 1
(a) A
differential equation
is an equation for an unknown function that contains the derivatives of that
unknown function. For example
y
00
(
t
) +
y
(
t
) = 0 is a differential equation for the unknown function
y
(
t
).
(b) A differential equation is called an
ordinary differential equation
(often shortened to “ODE”) if
only ordinary derivatives appear. That is, if the unknown function has only a single independent variable.
A differential equation is called a
partial differential equation
(often shortened to “PDE”) if partial
derivatives appear. That is, if the unknown function has more than one independent variable. For example
y
00
(
t
) +
y
(
t
) = 0 is an ODE while
∂
2
u
∂t
2
(
x, t
) =
c
2
∂
2
u
∂x
2
(
x, t
) is a PDE.
(c) The
order
of a differential equation is the order of the highest derivative that appears.
For example
y
00
(
t
) +
y
(
t
) = 0 is a second order ODE.
(d) An ordinary differential equation that is of the form
a
0
(
t
)
y
(
n
)
(
t
) +
a
1
(
t
)
y
(
n

1)
(
t
) +
· · ·
+
a
n

1
(
t
)
y
0
(
t
) +
a
n
(
t
)
y
(
t
) =
F
(
t
)
(1)
with given coefficient functions
a
0
(
t
),
· · ·
,
a
n
(
t
) and
F
(
t
) is said to be
linear
. Otherwise, the ODE is said
to be
nonlinear
. For example,
y
0
(
t
)
2
+
y
(
t
) = 0,
y
0
(
t
)
y
00
(
t
) +
y
(
t
) = 0 and
y
0
(
t
) =
e
y
(
t
)
are all nonlinear.
(e) The ODE (1) is said to have
constant coefficients
if the coefficients
a
0
(
t
),
a
1
(
t
),
· · ·
,
a
n
(
t
) are all
contants. Otherwise, it is said to have
variable coefficients
. For example, the ODE
y
00
(
t
) + 7
y
(
t
) = sin
t
is constant coefficient, while
y
00
(
t
) +
ty
(
t
) = sin
t
is variable coefficient.
(f) The ODE (1) is said to be
homogeneous
if
F
(
t
) is identically zero.
Otherwise, it is said to be
in
homogeneous
or
nonhomogeneous
.
For example, the ODE
y
00
(
t
) + 7
y
(
t
) = 0 is homogeneous, while
y
00
(
t
) + 7
y
(
t
) = sin
t
is inhomogeneous. A homogeneous ODE always has the trivial solution
y
(
t
) = 0.
(g) An
initial value problem
is a problem in which one is to find an unknown function
y
(
t
) that satisfies
both a given ODE and given initial conditions, like
y
(0) = 1,
y
0
(0) = 0.
(h) A
boundary value problem
is a problem in which one is to find an unknown function
y
(
t
) that satisfies
both a given ODE and given boundary conditions, like
y
(0) = 0,
y
(1) = 0.
Theorem 2
Assume that the coefficients
a
0
(
t
)
,
a
1
(
t
)
,
· · ·
,
a
n

1
(
t
)
,
a
n
(
t
)
and
F
(
t
)
are reasonably smooth,
bounded functions and that
a
0
(
t
)
is not zero.
(a) The general solution to the ODE (1) is of the form
y
(
t
) =
y
p
(
t
) +
C
1
y
1
(
t
) +
C
2
y
2
(
t
) +
· · ·
+
C
n
y
n
(
t
)
where
◦
n
is the order of the ODE (1)
◦
the particular solution,
y
p
(
t
)
, is any solution to (1)
◦
C
1
,
C
2
,
· · ·
,
C
n
are arbitrary constants
◦
y
1
,
y
2
,
· · ·
,
y
n
are
n
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 Spring '10
 YILMAZ
 Differential Equations, Equations, Derivative, Boundary value problem, ORDINARY DIFFERENTIAL EQUATIONS, Review of Ordinary Differential Equations

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