Undetermined Coefficient Annihilator Operator Approach
Recall
1. That a nonhomogeneous linear differential equation of order _{ } is an equation of the form
_{ }
The following differential equation is called the associated homogeneous equation _{ }
The coefficients _{ }can be functions of_{ }. However, we will discuss equations with constant coefficients.
2. That to obtain the general solution of a nonhomogeneous linear differential equation we must find:

The complementary function_{ }, which is general solution of the associated homogeneous differential equation.

Any particular solution _{ }of the nonhomogeneous differential equation.
3. That the general solution of the nonhomogeneous linear differential equation is given by
General Solution = Complementary Function + Particular Integral

Finding the complementary function has been completely discussed in an earlier lecture

In the previous lecture, we studied a method for finding particular integral of the nonhomogeneous equations. This was the method of undetermined coefficients developed from the viewpoint of superposition principle.

In the present lecture, we will learn to find particular integral of the nonhomogeneous equations by the same method utilizing the concept of differential annihilator operators.

In calculus, the differential coefficient _{ } is often denoted by the capital letter_{ }. So that
_{ }
The symbol _{ } is known as differential operator.

This operator transforms a differentiable function into another function, e.g.
_{ }

The differential operator _{ } possesses the property of linearity. This means that if _{ }are two differentiable functions, then
_{ }
Where _{ } and _{ } are constants. Because of this property, we say that _{ } is a linear differential operator.

Higher order derivatives can be expressed in terms of the operator _{ } in a natural manner:
_{ }
Similarly
_{ }_{ }

The following polynomial expression of degree _{ } involving the operator _{ }
_{ }
is also a linear differential operator.
For example, the following expressions are all linear differential operators
_{ }, _{ }, _{ }

Differential Equation in Terms of D
Any linear differential equation can be expressed in terms of the notation_{ }. Consider a 2^{nd} order equation with constant coefficients
_{ }
Since _{ }
Therefore the equation can be written as
_{ }
or _{ }
Now, we define another differential operator _{ }as
_{ }
Then the equation can be compactly written as
_{ }
The operator _{ } is a secondorder linear differential operator with constant coefficients.
Example 1
Consider the differential equation
_{ }
Since _{ }
Therefore, the equation can be written as
_{ }
Now, we define the operator _{ }as
_{ }
Then the given differential can be compactly written as
_{ }
Factorization of a differential operator

An nthorder linear differential operator
_{ }
with constant coefficients can be factorized, whenever the characteristics polynomial equation
_{ }
can be factorized.

The factors of a linear differential operator with constant coefficients commute.
Example 2
(a) Consider the following 2^{nd} order linear differential operator
_{ }
If we treat _{ } as an algebraic quantity, then the operator can be factorized as
_{ }
(b) To illustrate the commutative property of the factors, we consider a twicedifferentiable function_{ }. Then we can write
_{ }
To verify this we let
_{ }
Then
_{ }
or _{ }
or _{ }
or _{ }
Similarly if we let
_{ }
Then _{ }
or _{ }
or _{ }
Therefore, we can write from the two expressions that
_{ }
Hence _{ }

The operator _{ } can be factorized as
_{ }
or _{ }

The operator _{ } does not factor with real numbers.
The differential equation
_{ }
can be written as
_{ }
or _{ }
or _{ }
Annihilator Operator
Suppose that

L is a linear differential operator with constant coefficients.

y = f(x) defines a sufficiently differentiable function.

The function f is such that L(y)=0
Then the differential operator L is said to be an annihilator operator of the function f.
Example 5
Since
_{ } _{ } _{ } _{ }
Therefore, the differential operators
_{ }, _{ }, _{ }, _{ }
are annihilator operators of the following functions
_{ }
In general, the differential operator _{ } annihilates each of the functions
_{ }
Hence, we conclude that the polynomial function
_{ }
can be annihilated by finding an operator that annihilates the highest power of _{ }
Find a differential operator that annihilates the polynomial function
_{ }.
Solution
Since _{ }
Therefore _{ }
Hence, _{ } is the differential operator that annihilates the function_{ }
Note that the functions that are annihilated by an nthorder linear differential operator _{ } are simply those functions that can be obtained from the general solution of the homogeneous differential equation
_{ }
Example 7
Consider the homogeneous linear differential equation of order _{ }
_{ }
The auxiliary equation of the differential equation is
_{ }
_{ } _{ }
Therefore, the auxiliary equation has a real root _{ }of multiplicity_{ }. So that the differential equation has the following linearly independent solutions:
_{ }
Therefore, the general solution of the differential equation is
_{ }
So that the differential operator
_{ }
annihilates each of the functions
_{ }
Hence, as a consequence of the fact that the differentiation can be performed term by term, the differential operator
_{ }
annihilates the function
_{ }
Find an annihilator operator for the functions
(a) _{ }
(b) _{ }
Solution
(a) Since
_{ }
Therefore, the annihilator operator of function _{ } is given by
_{ }
We notice that in this case_{ }.
(b) Similarly
_{ }
or _{ }
or _{ }
Therefore, the annihilator operator of the function _{ }is given by
_{ }
We notice that in this case _{ }.
Example 9
Consider the differential equation
_{ }
The auxiliary equation is
_{ }
_{ } _{ }
Therefore, when _{ } are real numbers, we have from the quadratic formula
_{ }
Therefore, the auxiliary equation has the following two complex roots of multiplicity _{ }
_{ }
Thus, the general solution of the differential equation is a linear combination of the following linearly independent solutions
_{ }
_{ }
Hence, the differential operator
_{ }
is the annihilator operator of the functions
_{ }
_{ }
Example 10
If we take
_{ }
Then the differential operator
_{ }
becomes _{ }.
Also, it can be verified that
_{ } _{ }
_{ }
Therefore, the linear differential operator
_{ }
annihilates the functions
_{ }
Now, consider the differential equation
_{ }
The auxiliary equation is
_{ }
Therefore, the functions
_{ }
are the two linearly independent solutions of the differential equation
_{ },
Therefore, the operator also annihilates a linear combination of_{ } and_{ }, e.g.
_{ }.
Example 11
If we take
_{ }
Then the differential operator
_{ }
becomes
_{ }
Also, it can be verified that
_{ } _{ }
_{ }
and
_{ }
_{ }
Therefore, the linear differential operator
_{ }
annihilates the functions
_{ }
Example 12
Taking_{ }, the operator
_{ }
becomes
_{ }
Since _{ }
_{ }
Therefore, the differential operator annihilates the functions
_{ }
Note that

If a linear differential operator with constant coefficients is such that
_{ }, _{ }
i.e. the operator_{ }annihilates the functions _{ }and _{ }. Then the operator_{ } annihilates their linear combination.
_{ }.
This result follows from the linearity property of the differential operator_{ }.

Suppose that _{ }and _{ } are linear operators with constant coefficients such that
_{ }
and _{ }
then the product of these differential operators _{ } annihilates the linear sum
_{ }
So that _{ }
To demonstrate this fact we use the linearity property for writing
_{ }
Since _{ }
therefore _{ }
or _{ }
But we know that _{ }
Therefore _{ }
Example 13
Find a differential operator that annihilates the function
_{ }
Solution
Suppose that
_{ }
Then
_{ }
Therefore, _{ } annihilates the function _{ }
Example 14
Find a differential operator that annihilates the function
_{ }
Solution
Suppose that
_{ }
Then
_{ }
Therefore, the product of two operators
_{ }
annihilates the given function _{ }
Note that

The differential operator that annihilates a function is not unique. For example, _{ },
_{ }
_{ }
Therefore, there are 3 annihilator operators of the functions, namely _{ }, _{ }, _{ }

When we seek a differential annihilator for a function, we want the operator of lowest possible order that does the job.
Practice Exercises
Write the given differential equation in the form _{ }where _{ }is a differential operator with constant coefficients.

_{ }

_{ }

_{ }

_{ }
Factor the given differentiable operator, if possible.

_{ }

_{ }

_{ }

_{ }
Verify that the given differential operator annihilates the indicated functions

_{ }

_{ }
Find a differential operator that annihilates the given function.

_{ }

_{ }
Undetermined Coefficients: Annihilator Operator Approach
The method of undetermined coefficients that utilizes the concept of annihilator operator approach is also limited to nonhomogeneous linear differential equations
The form of _{ }:The input function _{ } has to have one of the following forms:

A constant function_{ }.

A polynomial function

An exponential function _{ }

The trigonometric functions _{ }

Finite sums and products of these functions.
Otherwise, we cannot apply the method of undetermined coefficients.
The Method
Consider the following nonhomogeneous linear differential equation with constant coefficients of order _{ }
_{ }
If _{ } denotes the following differential operator
_{ }
Then the nonhomogeneous linear differential equation of order _{ } can be written as
_{ }
The function _{ }should consist of finite sums and products of the proper kind of functions as already explained.
The method of undetermined coefficients, annihilator operator approach, for finding a particular integral of the nonhomogeneous equation consists of the following steps:
Step 1 Write the given nonhomogeneous linear differential equation in the form
_{ }
Step 2 Find the complementary solution _{ } by finding the general solution of the associated homogeneous differential equation:
_{ }
Step 3 Operate on both sides of the nonhomogeneous equation with a differential operator _{ } that annihilates the function g(x).
Step 4 Find the general solution of the higherorder homogeneous differential equation
_{ }
Step 5 Delete all those terms from the solution in step 4 that are duplicated in the complementary solution_{ }, found in step 2.
Step 6 Form a linear combination _{ } of the terms that remain. This is the form of a particular solution of the nonhomogeneous differential equation
_{ }
Step 7 Substitute _{ } found in step 6 into the given nonhomogeneous linear differential equation
_{ }
Match coefficients of various functions on each side of the equality and solve the resulting system of equations for the unknown coefficients in_{ }.
Step 8 With the particular integral found in step 7, form the general solution of the given differential equation as:
_{ }
Example 1
Solve _{ }.
Solution:
Step 1 Since _{ }
Therefore, the given differential equation can be written as
_{ }
Step 2 To find the complementary function_{ }, we consider the associated homogeneous differential equation
_{ }
The auxiliary equation is
_{ }
Therefore, the auxiliary equation has two distinct real roots.
_{ }, _{ },
Thus, the complementary function is given by
_{ }
Step 3 In this case the input function is
_{ }
Further _{ }
Therefore, the differential operator _{ }annihilates the function_{ }. Operating on both sides of the equation in step 1, we have
_{ }
This is the homogeneous equation of order 5. Next we solve this higher order equation.
Step 4 The auxiliary equation of the differential equation in step 3 is
_{ }
_{ }
_{ }
Thus its general solution of the differential equation must be
_{ }
Step 5 The following terms constitute _{ }
_{ }
Therefore, we remove these terms and the remaining terms are
_{ }
Step 6 This means that the basic structure of the particular solution _{ } is
_{ },
Where the constants_{ },_{ } and _{ } have been replaced, with A, B, and C, respectively.
Step 7 Since _{ }
_{ }
_{ }
Therefore _{ }
or _{ }
Substituting into the given differential equation, we have
_{ }
Equating the coefficients of _{ } and the constant terms, we have
_{ }
Solving these equations, we obtain
_{ }
Hence _{ }
Step 8 The general solution of the given nonhomogeneous differential equation is _{ }
_{ }.
Example 2
Solve _{ }
Solution:
Step 1 Since _{ }
Therefore, the given differential equation can be written as
_{ }
Step 2 We first consider the associated homogeneous differential equation to find_{ }
The auxiliary equation is
_{ }
Thus the auxiliary equation has real and distinct roots. So that we have
_{ }
Step 3 In this case the input function is given by
_{ }
Since _{ }
Therefore, the operators _{ } and _{ } annihilate _{ } and_{ }, respectively. So the operator _{ }annihilates the input function _{ } This means that
_{ }
We apply _{ } to both sides of the differential equation in step 1 to obtain
_{ }.
This is homogeneous differential equation of order 5.
Step 4 The auxiliary equation of the higher order equation found in step 3 is
_{ }
_{ }
_{ }
Thus, the general solution of the differential equation
_{ }
Step 5 First two terms in this solution are already present in _{ }
_{ }
Therefore, we eliminate these terms. The remaining terms are
_{ }
Step 6 Therefore, the basic structure of the particular solution _{ }must be
_{ }
The constants _{ }and _{ }have been replaced with the constants _{ }and_{ }, respectively.
Step 7 Since _{ }
Therefore _{ }
Substituting into the given differential equation, we have
_{ }.
Equating coefficients of _{ } and_{ }, we obtain
_{ }
Solving these equations we obtain
_{ }
_{ }.
Step 8 The general solution of the differential equation is then
_{ }.
Example 3
Solve _{ } .
Solution:
Step 1 The given differential equation can be written as
_{ }
Step 2 The associated homogeneous differential equation is
_{ }
Roots of the auxiliary equation are complex
_{ }
Therefore, the complementary function is
_{ }
Step 3 Since _{ }
Therefore the operators _{ } and _{ }annihilate the functions _{ } and_{ }. We apply _{ } to the nonhomogeneous differential equation
_{ }.
This is a homogeneous differential equation of order 5.
Step 4 The auxiliary equation of this differential equation is
_{ }
Therefore, the general solution of this equation must be
_{ }
Step 5 Since the following terms are already present in _{ }
_{ }
Thus we remove these terms. The remaining ones are
_{ }
Step 6 The basic form of the particular solution of the equation is
_{ } _{ }
The constants _{ }and _{ }have been replaced with _{ }and_{ }.
Step 7 Since _{ }
Therefore _{ }
Substituting in the given differential equation, we have
_{ }
Equating coefficients of _{ }and the constant terms, we have
_{ }
Thus _{ }
Step 8 Hence, the general solution of the given differential equation is
_{ }
or _{ }.
Example 4
Solve _{ }
Solution:
Step 1 The given differential equation can be written as
_{ }
Step 2 Consider the associated differential equation
_{ }
The auxiliary equation is
_{ } _{ }
Therefore _{ }
Step 3 Since _{ }
_{ }
Therefore, the operator _{ }annihilates the input function
_{ }
Thus operating on both sides of the nonhomogeneous equation with_{ }, we have
_{ }
or _{ }
This is a homogeneous equation of order 6.
Step 4 The auxiliary equation of this higher order differential equation is
_{ }
Therefore, the auxiliary equation has complex roots_{ }, and _{ }both of multiplicity 3. We conclude that
_{ }
Step 5 Since first two terms in the above solution are already present in _{ }
_{ }
Therefore, we remove these terms.
Step 6 The basic form of the particular solution is
_{ }
Step 7 Since _{ }
Therefore
_{ }
Substituting in the given differential equation, we obtain
_{ }
Equating coefficients of _{ } and _{ }, we obtain
_{ }
Solving these equations we obtain
_{ }
Thus _{ }
Step 8 Hence the general solution of the differential equation is
_{ }.
Example 5
Determine the form of a particular solution for
_{ }
Solution
Step 1 The given differential equation can be written as
_{ }
Step 2 To find the complementary function, we consider
_{ }
The auxiliary equation is
_{ }_{ }_{ }
The complementary function for the given equation is
_{ }
Step 3 Since _{ }
Applying the operator _{ } to both sides of the equation, we have
_{ }
This is homogeneous differential equation of order 4.
Step 4 The auxiliary equation is
_{ }
Therefore, general solution of the 4^{th} order homogeneous equation is
_{ }
Step 5 Since the terms _{ } are already present in_{ }, therefore, we remove these and the remaining terms are _{ }
Step 6 Therefore, the form of the particular solution of the nonhomogeneous equation is
_{ } _{ }
Note that the steps 7 and 8 are not needed, as we don’t have to solve the given differential equation.
Example 6
Determine the form of a particular solution for
_{ }.
Solution:
Step 1 The given differential can be rewritten as
_{ }
Step 2 To find the complementary function, we consider the equation
_{ }
The auxiliary equation is
_{ }
_{ }
_{ }
Thus the complementary function is
_{ }
Step 3 Since _{ }
Further _{ }
_{ }
_{ }
Therefore the following operator must annihilate the input function_{ }. Therefore, applying the operator _{ } to both sides of the nonhomogeneous equation, we have
_{ }
or _{ }
This is homogeneous differential equation of order 10.
Step 4 The auxiliary equation for the 10^{th} order differential equation is
_{ }
Hence the general solution of the 10^{th} order equation is
_{ }
Step 5 Since the following terms constitute the complementary function_{ }, we remove these _{ }
Thus the remaining terms are
_{ }
Hence, the form of the particular solution of the given equation is
_{ }
Practice Exercise
Solve the given differential equation by the undetermined coefficients.

_{ }

_{ }

_{ }

_{ }

_{ }

_{ }

_{ }

_{ }, _{ }, _{ }

_{ }, _{ }y(0)=2, _{ }, _{ }

_{ }, _{ }y(0)=0, _{ }, _{ }, _{ }
