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SYSTEMS OF LINEAR DIFFERENTIAL EQUATIONS

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)

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