SYSTEMS OF LINEAR DIFFERENTIAL EQUATIONS
SYSTEMS OF LINEAR DIFFERENTIAL EQUATIONS
A general system of linear 1
st
order of equations of the form
Where y and x are the dependent variables and t is the independent variable. A pair of
functions and is to be a solution of if and satisfies all equations in (1)
simultaneously.
Differential operator
The differential operator D defined as
where is an independent variable e.g. if is a
function of which is n times differentiable with respect to t, then
Using the D operator the equation 032
2
2
x
dt
dx
dt
xd
can be written as:
Note
The linear expression
In terms of D can be written as
Definition
The expression
is said to be linear operator of order n if
are all equal to a constant, then this expression is a linear operator of order n with
constant coefficients.
Usually we denote a linear operator by L, example:
is a linear operator of order 2
Properties of linear operators
Let and be two differentiable functions of and be a linear operator, then;
1
i)
Where c1 and c2 are constants
Example
If
Then
ii) Let
And be a function of which is times differentiable with respect to t,
where L is the operator of .
Example
Given that
Show that where is the product operator of
Solution
Clearly considering above
OPERATOR METHOD OF SOLVING SYSTEMS OF DIFFERENTIAL EQUATIONS OF
ORDER 1
Example
Find the general solution of the system
Solution
The system in differential operator form can be written as
Multiplying by and by we have
Adding and gives
The homogenous equation corresponding to this is
Whose characteristic equation is:
With the solutions:
Hence:
Using the method of undetermined coefficients:
Let
Substituting in the original ODE we have
thus:
Recall,
and satisfies the given system for all arbitrary constants. If the number of these constants in
both and equal to the order of the determinant of the matrix:
.
To find the relationship between the arbitrary constants we substitute and in the original
system
Substituting in
Alternative method
In this method we find the solution of one of the variables then eliminate the derivatives of the other
variable from the system to obtain a linear equation for the 2
nd
variable and the other variable together
with its derivatives. To illustrate this method consider the given system;
If we eliminate y and its derivatives from this system we have;
Where the solution is;
…………*
Adding and give
Exercise
Find the general solutions for each of the following systems
i)
ii)
iii)
