Basic Theory of Linear O.D.E.’s
Definition
: A
linear
O.D.E. of order n, in the independent variable x and the dependent
variable y is an equation that can be expressed in the form:
a
0
(x)y
n
( )
+
a
1
(x)y
n
!
1
(
)
+
.....
+
a
n
!
1
(x)
"
y
+
a
n
(x)y
=
b(x)
We will assume that a
0
(x), a
1
(x), …. , a
n
(x) and b(x) are continuous real-valued functions
on the interval [a, b] and a
0
(x)
≠
0 for at least one x in [a, b].
Remark
: A linear ODE satisfies the following conditions:
1- The dependent variable y and its derivatives occur to the first degree only.
2- No products of y and/or any of its derivatives appear in the equation.
3- No transcendental functions of y and/or its derivatives occur.
Definition
: Equations that not satisfy the above conditions are called nonlinear.
Remark
:
1) The functions a
0
(x), a
1
(x), …. , a
n
(x) are called the
coefficients
of the equation.
2) The right-hand member b(x) is called the
non-homogeneous
term.
3) If b(x) = 0, then the equation reduces to
a
0
(x)y
n
( )
+
a
1
(x)y
n
!
1
(
)
+
.....
+
a
n
!
1
(x)
"
y
+
a
n
(x)y
=
0
and it is called a
homogeneous linear differential equation (H.L.D.E.)
.
Example
:
1)
d
3
y
dx
3
!
cos(x)
d
2
y
dx
2
+
3x
dy
dx
+
x
3
y
=
e
x
is a linear differential equation of order 3,
where a
0
(x) = 1, a
2
(x) = - cos(x), a
3
(x) = 3x, a
4
(x) = x
3
and b(x) = e
x
, they are all
continuous function on
!
.
Definition
: If a
0
, a
1
, …. , a
n
are all constants, then we have a linear differential equation
with constant coefficients, otherwise we have an equation with variable coefficients.
Example:
1) 3y’’’ + 2y’’ – 4y’ – 7y = sin x is an equation with constant coefficients.
2) 2y’’ + xy’ – e
x
y = 0 is an equation with variable coefficients.
Theorem
: Let
a
0
(x)y
n
( )
+
a
1
(x)y
n
!
1
(
)
+
.....
+
a
n
!
1
(x)
"
y
+
a
n
(x)y
=
b(x)
with a
0
(x), a
1
(x), …. , a
n
(x) and b(x) are continuous real-valued functions on the interval
[a, b] and a
0
(x)
≠
0 for at least one x in [a, b].
Let x
0
be a point on the interval [a, b] and let c
0
, c
1
, c
n-1
be n arbitrary constants.
Conclusion
:
There exists a unique function f(x) defined on [a, b] such that f(x) is a solution of the
linear differential equation and satisfies
f(x
0
) = c
0
, f’(x
0
) = c
1
, f’’(x
0
) = c
2
, … , f
(n-1)
(x
0
) = c
n-1
.