Engineering Mathematics Term Paper Review

7/30/2019 Engineering Mathematics Term Paper Review

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TERM PAPER

OFENGINEERING MATHAMETICS-I

MTH101

Topic: Write about the method of undetermined

coefficients & method of variation of parameters. Discuss

& compare the advantages and disadvantages of eachmethod.

Illustrate your findings with examples

Sub. To: Sub. By:

Mr. Bharpur Singh Name: - Vaibhav Kumar Tripathi

(Deptt. Of Mathematics) Roll no- RG4003A34

Program-B.Tech-M.Tech (ME)

Program code-1208-D

Reg.no-11000653

Section-G4003


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ACKNOWLEDGEMENT

I would like to thank all those who have encouraged me to Submit a

project related to Write about the method of undetermined

coefficients & method of variation of parameters. Discuss

& compare the advantages and disadvantages of each

method.

Illustrate your findings with examplesas a term paper.It is an extremely arduous job to acknowledge the invaluable help

rendered by all those venerated souls who directly or indirectlycontributes to make feat as possible for me. I certainly feel elated and

privileged to express my deep sense to gratitude to all of them.

The entire term paper would not have come to a state of fruition

without them. During term paper work, I learnt many things.

Everybody was extremely co-operative. I thank all the members of

LOVELY PROFESSIONAL UNIVERSITY who extended theirhands for help, co-operation and kindness whenever and wherever

required

VAIBHAV KUMAR TRIPATHI


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CONTENT TABLE

1. INTRODUCTION TO METHOD OF UNDETERMINED

COEFFICIENTS.

2. METHODOLOGY

3. TYPICAL FORM OF SOLUTION

4. EXAMPLES

5. INTRODUCTION TO METHOD OF VARIATIONS OF

PARAMERTERS

6. METHODOLOGY

7. EXAMPLES

8. ADVANTAGES AND DISADVANTAGES OF EACH

METHOD WITH EXAMPLES

9. REFERNCES


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INTRODUCTION TO METHOD OF UNDETERMINED

COEFFICIENTS

In Mathematics, the method of undetermined coefficients, otherwise

known as the Lucky Guess Method, is an approach to finding a

particular solution to a certain inhomogeneous ordinary differentialequations and recurrence relation. It is closely related to annihilator

method (variation of parameter method), but instead of using a

particular kind of differential operator (the annihilator) in order to find

the best possible from of the particular solution, a guess is made as to

the appropriate form , which is then tested by differentiating the

resulting equation. For complex equations, the annihilator method or

variation of parameters is less time consuming to perform.

METHODOLOGY

Considering a linear non-homogeneous ordinary differential equation

with the form

any(n) + a(n 1)y(n 1) + ... + a1y' + a0y = g(x)

The method consists of finding the homogeneous solution yc for the

complementary function and a particular solution yp based on g(x).

Then the general solution y to the equation would be

y = yc + yp

If g(x) consists of the sum of two functions h(x) + w(x) and we say that

yp1 is the solution based on h(x) and yp2 the solution based on w(x).


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Then, using the superposition principle, we can say that the particular

solution yp is

yp = yp1 + yp2

TYPICAL FORM OF SOLUTION

In order to find the particular solution, we need to 'guess' its form, with

some coefficients left as variables to be solved for. This takes the form of

the first derivative of complementary function. Below is a table of some

typical functions and the solution to guess for them.

Function of x Form for y

If a term in the above particular solution for y appears in the

homogeneous solution, it is necessary to multiply by a sufficiently large

power of x in order to make the two solutions linearly independent. If

the function of x is a sum of terms in the above table, the particular

solution can be guessed using a sum of the corresponding terms for y.

EXAMPLES

EXAMPLE1

Find a particular solution of the equation


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The right side t cos t has the form

With n=1, =0, and =1.

Since + i = i is a simple root of the characteristic equation

We should try a particular solution of the form

Substituting yp into the differential equation, we have the identity

Comparing both sides, we have


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Which has the solution A0 = 0, A1 = 1/4, B0 = 1/4, B1 = 0. We then have

a particular solution

Example 2

Consider the following linear inhomogeneous differential equation:

This is like the first example above, except that the inhomogeneous part

(ex) is not linearly independent to the general solution of the

homogeneous part (c1ex); as a result, we have to multiply our guess by a

sufficiently large power of x to make it linearly independent.

Here our guess becomes:

Yp= A x ex.

By substituting this function and its derivative into the differential

equation, one can solve for A:

Axex + Aex = Axex + ex

A = 1.

So, the general solution to this differential equation is thus:

y = c1ex + xex.


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Example 3

Find the general solution of the equation:

f(t), t2, is a polynomial of degree 2, so we look for a solution using the

same form,

yp = At2 + Bt + C, where

Plugging this particular solution with constants A, B, and C into the

original equation yields,

2At + B = t2 (At2 + Bt + C), where

T2 At2 = 0 and Bt = 2At and C = B

Replacing resulting constants,

Yp = t2 2t + 2

To solve for the general solution,y = yp + yh

Where yh is the homogeneous solution yh = c1e-1 therefore the general

solution is:

y = t2 2t + 2 + c1 e-t


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INTRODUCTION TO METHOD OF VARIATIONS OF

PARAMERTERS

In Mathematics, Variations of Parameters, also knownas variations ofconstant, is a general method to solve inhomogeneous linear ordinary

differential equations. It was developed by Joseph Louis Lagranges.

For first-order inhomogeneous linear differential equations its usually

possible to find solutions via integrating factor or undermined

coefficient with considerably less effort, although those methods are

rather heuristic that involve guessing and dont work for all

inhomogeneous linear differential equations.

Variations of parameters extend to linear partial differential equations

as well, specifically to inhomogeneous problems for linear evolution

equations like that heat equations, wave equations, and vibrating plate

equations. In this setting, the method is more often known as Duhamels

principle, named for Jean-Marie Duhamel who was first applied the

method to solve the inhomogeneous heat equations.


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METHODOLOGY

Given an ordinary non-homogeneous linear differential equation of

order n

(i)

Let be a fundamental system of the corresponding

homogeneous equation

(ii)

Then a particular solution to the non-homogeneous equation is given by

(iii)

Where the ci(x) are continuous functions which satisfy the equations

(iv)

(Results from substitution of (iii) into the homogeneous case (ii); )

And


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(v)

(Results from substitution of (iii) into (i) and applying (iv);

Ci'(x) = 0 for all x and i is the only way to satisfy the condition, since allyi(x) are linearly independent. It implies that all ci(x) are independent of

x in the homogeneous case b(x)=0. )

This linear system of n equations can then be solved using Cramer's

rule yielding

Where W(x) is the Wronskian determinant of the fundamental system

and Wi(x) is the Wronskian determinant of the fundamental system

with the i-th column replaced by

The particular solution to the non-homogeneous equation can then be

written as


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EXAMPLES

SPECIFIC SECOND ORDER EQUATION

Let us solve

We want to find the general solution to the differential equation, that is,

we want to find solutions to the homogeneous differential equation

From the characteristic equation

Since we have a repeated root, we have to introduce a factor of x for one

solution to ensure linear independence.

So, we obtain u1 = e2x, and u2 = xe2x. The Wronskian of these two

functions is


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Because the Wronskian is non-zero, the two functions are linearly

independent, so this is in fact the general solution for the homogeneous

differential equation (and not a mere subset of it).

We seek functions A(x) and B(x) so A(x)u1 + B(x)u2 is a general solution

of the non-homogeneous equation. We need only calculate the integrals

That is,

Where C1 and C2 are constants of integration.


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General second order equation

We have a differential equation of the form

And we define the linear operator

Where D represents the differential operator. We therefore have to

solve the equation Lu(x) = f(x) for u(x), where L and f(x) are known.

We must solve first the corresponding homogeneous equation:

By the technique of our choice. Once we've obtained two linearly

independent solutions to this homogeneous differential equation

(because this ODE is second-order) call them u1 and u2 we can

proceed with variation of parameters.

Now, we seek the general solution to the differential equation uG(x)

which we assume to be of the form

Here, A(x) and B(x) are unknown and u1(x) and u2(x) are the solutions

to the homogeneous equation. Observe that if A(x) and B(x) are


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constants, then LuG(x) = 0. We desire A=A(x) and B=B(x) to be of the

form

Now,

And since we have required the above condition, then we have

Differentiating again (omitting intermediary steps)

Now we can write the action of L upon uG as

Since u1 and u2 are solutions, then

We have the system of equations

Expanding,

So the above system determines precisely the conditions

We seek A(x) and B(x) from these conditions, so, given


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We can solve for (A(x), B(x))T, so

Where W denotes the Wronskian of u1 and u2. (We know that W is

nonzero, from the assumption that u1 and u2 are linearly independent.)

So,

While homogeneous equations are relatively easy to solve, this method

allows the calculation of the coefficients of the general solution of the

inhomogeneous equation, and thus the complete general solution of the

inhomogeneous equation can be determined.

Note that A(x) and B(x) are each determined only up to an arbitrary

additive constant (the constant of integration); one would expect two

constants of integration because the original equation was second order.

Adding a constant to A(x) or B(x) does not change the value of LuG(x)

because L is linear.


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ADVANTAGES AND DISADVANTAGES OF EACH METHOD

WITH EXAMPLES.

Undetermined Coefficients

Advantages:

a. Easy mathematics: involves NO integration.

b. Can be used to find a particular solution without having to find

eigenvectors.

Disadvantages:

a. Only works when g contains nice functions (polynomials,

exponentials, and trigs).

b. P must have constant coefficients.

Variation of Parameters

Advantages:

a. May be applied to solve any problem.


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b. Easy to remember: the only formulae you need are are u0 = g

And xP= u.

Disadvantages:

a. Solving equations can get ugly because each component of the

vectors involved can contain a complicated expression and row-

reduction becomes cumbersome.

b. Integrating equationto find solution can sometimes be difficult.

c. Doing the matrix multiplication of equations can be tedious.

REFERENCES

1. Boyce, W. E.; DiPrima, R. C. (1965). Elementary Differential

Equations and Boundary Value problems. Wiley Interscience.

2. ^ Kenneth H. Rosen: Handbook of Discrete and Combinatorial

Mathematics. CRC Press 2000. (3.3.3 Nonhomogeneous

Recurrence Relations)

3. ^ Kenneth H. Rosen: Handbook of Discrete and Combinatorial

Mathematics. CRC Press 2000. (3.3.3 Nonhomogeneous

Recurrence Relations)

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