RB1901B34MTH

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lovely professional university

Name: kapish grover

Section: b1901

Roll no: b34

Course: mth102

Topic: method of undetermined coefficients &

method of variation of parameters

Submitted to: geeta m kodabagi

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ACKNOWLEDGMENT

I thank and would like to pay my respect to my respected Maths teacher Mrs.

Geeta M Kodabagi from the depth of my heart for the guidance and the help

which he has provided me in order to complete my term paper successfully by

providing me with adequate information and teaching the very basic concepts

about the topic which was very helpful to me in achieving my goal.

I am also very thankful to all those persons who have helped me in any slightest

manner in completing my term paper successfully.

I would also like to thank GOD for providing me with strength to successfully

complete my term paper solely.

And I declare that this term paper is my own work and none of its material is taken

from any other source except from those which are mentioned.

Submitted by:

KAPISH GROVER

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TOPIC IN DETAIL: Write about the method of undetermined

coefficients & method of variation of parameters. Discuss &

compare the advantages & disadvantages of each method.Illustrate your findings with examples.

ANS:

[A.] Method of Undetermined Coefficient

This method is based on a guessing technique. That is, we will guess the form of and then plug

it in the equation to find it. However, it works only under the following two conditions:

Condition 1: the associated homogeneous equations has constant coefficients;

Condition 2: the nonhomogeneous termg(x) is a special form

whereP(x) andL(x) are polynomial functions.

Note that we may assume thatg(x) is a sum of such functions (see the remark below formore on this).

Assume that the two conditions are satisfied. Consider the equation

where a, b and c are constants and

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where is a polynomial function with degree n. Then a particular solution is given by

where

,

where the constants and have to be determined. The powers is equal to 0 if is not a

root of the characteristic equation. If is a simple root, thens=1 ands=2 if it is a doubleroot.

Remark:

If the nonhomogeneous termg(x) satisfies the following

where are of the forms cited above, then we split the original equation intoNequations

then find a particular solution . A particular solution to the original equation is given by

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Summary:

Let us summarize the steps to follow in applying this method:

(1)

First, check that the two conditions are satisfied;

(2)If the equation is given as

,

where or , where is

a polynomial function with degree n, then split this equation into N equations

;

(3)

Write down the characteristic equation , and find its roots;

(4)

Write down the number . Compare this number to the roots of the

characteristic equation found in previous step.

(4.1)

If is not one of the roots, then set s = 0;

(4.2)

If is one of the two distinct roots, set s = 1;

(4.3)

If is equal to both root (which means that the characteristic equation has

a double root), set s=2.

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In other words, s measures how many times is a root of the characteristic

equation;

(5)

Write down the form of the particular solution

where

(6)

Find the constants and by plugging into the equation

(7)

Once all the particular solutions are found, then the particular solution of the

original equation is

EXAMPLES:

Example1

Find a particular solution to the equation

Solution: Let us follow these steps:

(1)

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First, we notice that the conditions are satisfied to invoke the method of

undetermined coefficients.

(2)We split the equation into the following three equations:

(3)

The root of the characteristic equation are r=-1 and r=4.

(4.1)

Particular solution to Equation (1):

Since , and , then , which is not one of the roots. Thens=0.The particular solution is given as

If we plug it into the equation (1), we get

,

which impliesA = -1/2, that is,

(4.2)


Since , and , then , which is not one of the roots. Thens=0.

The particular solution is given as


,

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which implies

Easy calculations give , and , that is

(4.3)


Since , and , then which is one of the roots. Thens=1.



,

which implies , that is

(5)

A particular solution to the original equation is

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Example 2

Find a particular solution of

Solution:

Here with and Also, the characteristic polynomial,

Note that is not a root of Thus, we assume This on

substitution gives

So, we choose which gives a particular solution as

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[B.] Method of Variation of Parameters

This method has no prior conditions to be satisfied.

Consider the equation

In order to use the method of variation of parameters we need to know that is a set offundamental solutions of the associated homogeneous equationy'' +p(x)y' + q(x)y = 0. We know

that, in this case, the general solution of the associated homogeneous equation is

. The idea behind the method of variation of parameters is to look for aparticular solution such as

where and are functions. From this, the method got its name.

The functions and are solutions to the system

,

which implies

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,

where is the wronskian of and . Therefore, we have

Summary:

Let us summarize the steps to follow in applying this method:

(1)

Find a set of fundamental solutions of the associated homogeneous equationy'' + p(x)y' + q(x)y = 0.

;

(2)

Write down the form of the particular solution

(3)

Write down the system

;

(4)

Solve it. That is, find and ;

(5)

Plug and into the equation giving the particular solution.
http://www.sosmath.com/diffeq/second/linearind/linearind.htmlhttp://www.sosmath.com/diffeq/second/linearind/linearind.html

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Example:

Find the particular solution to

Solution: Let us follow the steps:

(1)

A set of fundamental solutions of the equationy'' +y = 0 is ;

(2)


(3)

We have the system

;

(4)

We solve for and , and get

Using techniques of integration, we get

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;

(5)

The particular solution is:

,

or

Remark:

Note that since the equation is linear, we may still split if necessary. For example, we may split

the equation

,

into the two equations

then, find the particular solutions for (1) and for (2), to generate a particular solution for the

original equation by

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There are no restrictions on the method to be used to find or . For example, we can use the

method of undetermined coefficients to find , while for , we are only left with the variationof parameters.

[C.]Discuss & compare the advantages &

disadvantages of each method

Advantage of method of undetermined coefficients:

The method of undetermined coefficients is a way of finding

an exact solution when a guess can be made as to the general form of the solution.

One of the main advantages of this method is that it reduces the problem

down to an algebra problem. The algebra can get messy on occasion, but for most of theproblems it will not be terribly difficult. Another nice thing about this method is that the

complimentary solution will not be explicitly required, although as we will see knowledge of the

complimentary solution will be needed in some cases and so well generally find that as well.

The method is quite simple. All that we need to do is look atg(t) and make a guess as

to the form ofYP(t) leaving the coefficient(s) undetermined (and hence the name of the method).

Plug the guess into the differential equation and see if we can determine values of the

coefficients. If we can determine values for the coefficients then we guessed correctly, if we

cant find values for the coefficients then we guessed incorrectly.
http://www.sosmath.com/diffeq/second/guessing/guessing.htmlhttp://planetmath.org/encyclopedia/MultipleRoot.htmlhttp://www.sosmath.com/diffeq/second/guessing/guessing.htmlhttp://planetmath.org/encyclopedia/MultipleRoot.html

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Disadvantage of method of undetermined coefficients:

There are two disadvantages to this method. First, it will

only work for a fairly small class ofg(t)s. The class ofg(t)s for which the methodworks, does include some of the more common functions, however, there are many

functions out there for which undetermined coefficients simply wont work. Second,

it is generally only useful for constant coefficient differential equations.

Advantage of variation of parameters:

The method of Variation of Parameters is a much more general method that can be

used in many more cases.

The method of variation of parameters can also be used in linear differential equationswith variable coefficients. However, the complementary solution must be found first and

sometimes the final solution can not be obtained without numerical integration.

We will see that variation of paramiters method depends on integration

while the Method of Undetermined Coefficients is purely algebraic which, for some at least, is anadvantage.

Disadvantage of variation of parameters:

There are two disadvantages to the method. First, the complimentary solution is absolutely

required to do the problem. This is in contrast to the method of undetermined coefficients where

it was advisable to have the complimentary solution on hand, but was not required. Second, as

we will see, in order to complete the method we will be doing a couple of integrals and there is

no guarantee that we will be able to do the integrals. So, while it will always be possible to write

down a formula to get the particular solution, we may not be able to actually find it if theintegrals are too difficult or if we are unable to find the complimentary solution.

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COMAPARISON OF THE METHODS:

The method of undetermined coefficients can be used to find a particular solutionyp of a

nonhomogeneous linear d.e. if the d.e. has constant coefficients and the nonhomogeneous term is

a polynomial, an exponential, a sine or a cosine, or a sum or product of these.

If the d.e. has variable coefficients and/or the nonhomogeneous term is

something other than a polynomial, exponential, sine, cosine, or sum or product of these, youmust use another method (e.g. variation of parameters).

The method of undetermined coefficients is applicable only if

the RHS of the non-homogeneous equation is itself a particular solution of

some homogeneous linear differential equation with constant coefficients.

Since Sec(2t) is not such a solution, this method is not applicable.

Method of variation may sound more general than the method of

undetermined coefficients. We will see that this method depends on

integration while the previous one is purely algebraic which, for some at

least, is an advantage.

We will see that variation of paramiters method depends on integration while the Method of

Undetermined Coefficients is purely algebraic.

[D.]Bibliography:

1. http://tutorial.math.lamar.edu/Classes/DE/VariationofParameters.aspx

2. http://www.efunda.com/math/ode/linearode_varypar.cfm
http://tutorial.math.lamar.edu/Classes/DE/VariationofParameters.aspxhttp://www.efunda.com/math/ode/linearode_varypar.cfmhttp://tutorial.math.lamar.edu/Classes/DE/VariationofParameters.aspxhttp://www.efunda.com/math/ode/linearode_varypar.cfm

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3. http://www.cliffsnotes.com/study_guide/Variation-of-

Parameters.topicArticleId-19736,articleId-19722.html

4. http://www.cliffsnotes.com/study_guide/The-Method-of-Undetermined-

Coefficients.topicArticleId-19736,articleId-19721.html

5. http://books.google.co.in/books?id=XyG-VC_-

5G4C&pg=PT438&lpg=PT438&dq=method+of+undetermined+

coefficients+

%26+method+of+variation+of+parameters&source=bl&ots=0

gK8oiFt88&sig=yysgIuB5tz4yPkYmYn1DvMF5W4U&hl=en&ei=

9HLgS7C1B8y9rAexiaHyBg&sa=X&oi=book_result&ct=result&r

esnum=9&ved=0CDIQ6AEwCA#v=onepage&q=method%20of

%20undetermined%20coefficients%20%26%20method%20of

%20variation%20of%20parameters&f=false
http://www.cliffsnotes.com/study_guide/Variation-of-Parameters.topicArticleId-19736,articleId-19722.htmlhttp://www.cliffsnotes.com/study_guide/Variation-of-Parameters.topicArticleId-19736,articleId-19722.htmlhttp://www.cliffsnotes.com/study_guide/The-Method-of-Undetermined-Coefficients.topicArticleId-19736,articleId-19721.htmlhttp://www.cliffsnotes.com/study_guide/The-Method-of-Undetermined-Coefficients.topicArticleId-19736,articleId-19721.htmlhttp://books.google.co.in/books?id=XyG-VC_-5G4C&pg=PT438&lpg=PT438&dq=method+of+undetermined+coefficients+%26+method+of+variation+of+parameters&source=bl&ots=0gK8oiFt88&sig=yysgIuB5tz4yPkYmYn1DvMF5W4U&hl=en&ei=9HLgS7C1B8y9rAexiaHyBg&sa=X&oi=book_result&ct=result&resnum=9&ved=0CDIQ6AEwCA#v=onepage&q=method%20of%20undetermined%20coefficients%20%26%20method%20of%20variation%20of%20parameters&f=falsehttp://books.google.co.in/books?id=XyG-VC_-5G4C&pg=PT438&lpg=PT438&dq=method+of+undetermined+coefficients+%26+method+of+variation+of+parameters&source=bl&ots=0gK8oiFt88&sig=yysgIuB5tz4yPkYmYn1DvMF5W4U&hl=en&ei=9HLgS7C1B8y9rAexiaHyBg&sa=X&oi=book_result&ct=result&resnum=9&ved=0CDIQ6AEwCA#v=onepage&q=method%20of%20undetermined%20coefficients%20%26%20method%20of%20variation%20of%20parameters&f=falsehttp://books.google.co.in/books?id=XyG-VC_-5G4C&pg=PT438&lpg=PT438&dq=method+of+undetermined+coefficients+%26+method+of+variation+of+parameters&source=bl&ots=0gK8oiFt88&sig=yysgIuB5tz4yPkYmYn1DvMF5W4U&hl=en&ei=9HLgS7C1B8y9rAexiaHyBg&sa=X&oi=book_result&ct=result&resnum=9&ved=0CDIQ6AEwCA#v=onepage&q=method%20of%20undetermined%20coefficients%20%26%20method%20of%20variation%20of%20parameters&f=falsehttp://books.google.co.in/books?id=XyG-VC_-5G4C&pg=PT438&lpg=PT438&dq=method+of+undetermined+coefficients+%26+method+of+variation+of+parameters&source=bl&ots=0gK8oiFt88&sig=yysgIuB5tz4yPkYmYn1DvMF5W4U&hl=en&ei=9HLgS7C1B8y9rAexiaHyBg&sa=X&oi=book_result&ct=result&resnum=9&ved=0CDIQ6AEwCA#v=onepage&q=method%20of%20undetermined%20coefficients%20%26%20method%20of%20variation%20of%20parameters&f=falsehttp://books.google.co.in/books?id=XyG-VC_-5G4C&pg=PT438&lpg=PT438&dq=method+of+undetermined+coefficients+%26+method+of+variation+of+parameters&source=bl&ots=0gK8oiFt88&sig=yysgIuB5tz4yPkYmYn1DvMF5W4U&hl=en&ei=9HLgS7C1B8y9rAexiaHyBg&sa=X&oi=book_result&ct=result&resnum=9&ved=0CDIQ6AEwCA#v=onepage&q=method%20of%20undetermined%20coefficients%20%26%20method%20of%20variation%20of%20parameters&f=falsehttp://books.google.co.in/books?id=XyG-VC_-5G4C&pg=PT438&lpg=PT438&dq=method+of+undetermined+coefficients+%26+method+of+variation+of+parameters&source=bl&ots=0gK8oiFt88&sig=yysgIuB5tz4yPkYmYn1DvMF5W4U&hl=en&ei=9HLgS7C1B8y9rAexiaHyBg&sa=X&oi=book_result&ct=result&resnum=9&ved=0CDIQ6AEwCA#v=onepage&q=method%20of%20undetermined%20coefficients%20%26%20method%20of%20variation%20of%20parameters&f=falsehttp://books.google.co.in/books?id=XyG-VC_-5G4C&pg=PT438&lpg=PT438&dq=method+of+undetermined+coefficients+%26+method+of+variation+of+parameters&source=bl&ots=0gK8oiFt88&sig=yysgIuB5tz4yPkYmYn1DvMF5W4U&hl=en&ei=9HLgS7C1B8y9rAexiaHyBg&sa=X&oi=book_result&ct=result&resnum=9&ved=0CDIQ6AEwCA#v=onepage&q=method%20of%20undetermined%20coefficients%20%26%20method%20of%20variation%20of%20parameters&f=falsehttp://books.google.co.in/books?id=XyG-VC_-5G4C&pg=PT438&lpg=PT438&dq=method+of+undetermined+coefficients+%26+method+of+variation+of+parameters&source=bl&ots=0gK8oiFt88&sig=yysgIuB5tz4yPkYmYn1DvMF5W4U&hl=en&ei=9HLgS7C1B8y9rAexiaHyBg&sa=X&oi=book_result&ct=result&resnum=9&ved=0CDIQ6AEwCA#v=onepage&q=method%20of%20undetermined%20coefficients%20%26%20method%20of%20variation%20of%20parameters&f=falsehttp://books.google.co.in/books?id=XyG-VC_-5G4C&pg=PT438&lpg=PT438&dq=method+of+undetermined+coefficients+%26+method+of+variation+of+parameters&source=bl&ots=0gK8oiFt88&sig=yysgIuB5tz4yPkYmYn1DvMF5W4U&hl=en&ei=9HLgS7C1B8y9rAexiaHyBg&sa=X&oi=book_result&ct=result&resnum=9&ved=0CDIQ6AEwCA#v=onepage&q=method%20of%20undetermined%20coefficients%20%26%20method%20of%20variation%20of%20parameters&f=falsehttp://books.google.co.in/books?id=XyG-VC_-5G4C&pg=PT438&lpg=PT438&dq=method+of+undetermined+coefficients+%26+method+of+variation+of+parameters&source=bl&ots=0gK8oiFt88&sig=yysgIuB5tz4yPkYmYn1DvMF5W4U&hl=en&ei=9HLgS7C1B8y9rAexiaHyBg&sa=X&oi=book_result&ct=result&resnum=9&ved=0CDIQ6AEwCA#v=onepage&q=method%20of%20undetermined%20coefficients%20%26%20method%20of%20variation%20of%20parameters&f=falsehttp://www.cliffsnotes.com/study_guide/Variation-of-Parameters.topicArticleId-19736,articleId-19722.htmlhttp://www.cliffsnotes.com/study_guide/Variation-of-Parameters.topicArticleId-19736,articleId-19722.htmlhttp://www.cliffsnotes.com/study_guide/The-Method-of-Undetermined-Coefficients.topicArticleId-19736,articleId-19721.htmlhttp://www.cliffsnotes.com/study_guide/The-Method-of-Undetermined-Coefficients.topicArticleId-19736,articleId-19721.htmlhttp://books.google.co.in/books?id=XyG-VC_-5G4C&pg=PT438&lpg=PT438&dq=method+of+undetermined+coefficients+%26+method+of+variation+of+parameters&source=bl&ots=0gK8oiFt88&sig=yysgIuB5tz4yPkYmYn1DvMF5W4U&hl=en&ei=9HLgS7C1B8y9rAexiaHyBg&sa=X&oi=book_result&ct=result&resnum=9&ved=0CDIQ6AEwCA#v=onepage&q=method%20of%20undetermined%20coefficients%20%26%20method%20of%20variation%20of%20parameters&f=falsehttp://books.google.co.in/books?id=XyG-VC_-5G4C&pg=PT438&lpg=PT438&dq=method+of+undetermined+coefficients+%26+method+of+variation+of+parameters&source=bl&ots=0gK8oiFt88&sig=yysgIuB5tz4yPkYmYn1DvMF5W4U&hl=en&ei=9HLgS7C1B8y9rAexiaHyBg&sa=X&oi=book_result&ct=result&resnum=9&ved=0CDIQ6AEwCA#v=onepage&q=method%20of%20undetermined%20coefficients%20%26%20method%20of%20variation%20of%20parameters&f=falsehttp://books.google.co.in/books?id=XyG-VC_-5G4C&pg=PT438&lpg=PT438&dq=method+of+undetermined+coefficients+%26+method+of+variation+of+parameters&source=bl&ots=0gK8oiFt88&sig=yysgIuB5tz4yPkYmYn1DvMF5W4U&hl=en&ei=9HLgS7C1B8y9rAexiaHyBg&sa=X&oi=book_result&ct=result&resnum=9&ved=0CDIQ6AEwCA#v=onepage&q=method%20of%20undetermined%20coefficients%20%26%20method%20of%20variation%20of%20parameters&f=false

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