Undetermined Coefficient Annihilator Operator Approach Recall




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Lecture 12 Undetermined Coefficient - Annihilator Approach


Undetermined Coefficient Annihilator Operator Approach

Recall
1. That a non-homogeneous 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 non-homogeneous linear differential equation we must find:


  • The complementary function , which is general solution of the associated homogeneous differential equation.

  • Any particular solution of the non-homogeneous differential equation.

3. That the general solution of the non-homogeneous 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 non-homogeneous 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 non-homogeneous equations by the same method utilizing the concept of differential annihilator operators.


  • Differential 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 2nd 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 second-order 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 nth-order 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 2nd 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 twice-differentiable 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

  • Example 3


  1. The operator can be factorized as

or



  1. The operator does not factor with real numbers.



  • Example 4


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



  • Example 6


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 nth-order 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




  • Example 8


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 non-homogeneous 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 non-homogeneous linear differential equation with constant coefficients of order

If denotes the following differential operator



Then the non-homogeneous 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 non-homogeneous equation consists of the following steps:
Step 1 Write the given non-homogeneous 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 non-homogeneous equation with a differential operator that annihilates the function g(x).

Step 4 Find the general solution of the higher-order 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 non-homogeneous differential equation



Step 7 Substitute found in step 6 into the given non-homogeneous 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 non-homogeneous 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 non-homogeneous 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 non-homogeneous 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 4th 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 non-homogeneous 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 non-homogeneous equation, we have



or

This is homogeneous differential equation of order 10.
Step 4 The auxiliary equation for the 10th order differential equation is

Hence the general solution of the 10th 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.
















  1. , ,

  2. , y(0)=2, ,

  3. , y(0)=0, , ,





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