SERIES SOLUTION TO ORDINARY DIFFERENTIAL

EQUATIONS

-: CREATED BY :-

ALAY MEHTA 141080106011SHIVANI PATEL 141080106021KAVIN RAVAL 141080106026KUNTAL SONI 141080106028

ADVANCE ENGINEERING MATHEMATICS [A.E.M.]

INTRODUCTION

In many “ENGINEERING” applications, we come across the differential equations which are having coefficients.

So, for solving this types of problems we have different methods POWER SERIES METHOD. FROBENIOUS METHOD.

BASIC CONCEPTS 1) POWER SERISE:~ A series from

where b0,b1,…and x0 are constants(real or complex) and x varies around x0 is called a POWER SERISE in (x-x0) in one variable.In particular, when x0=0, then

It called power serise in x.

0

0 )(n

n xxbxf

......,2210

0

xbxbbxb n

nn

CONVERGENCE OF POWER SERISE As far as the convergence of power

series concern, we say that a power series converges,

For x=a:

and this series will converge if limit of partial sums

n

nn xab )( 0

0

n

nn

nxab )( 0

00lim

CONVERGENCE OF POWER SERISE There is some +ve number R such that the

series converges for|x-x0|<R and diverges for |x-x0|>R

o The number R is called radius of converges of the power series.

o If the series only converges at 0, then R is 0, If converges to every where then R is ∞.

o The collections of values of x for which the power series converge is called interval or range of convergence.

POWER SERIES SOLUTION THEOREM

If x=x0 is ordinary of differential equation

where ,

is obtained as linear combination of two linearly independent power series solutions y1 and y2, each of which is of the from and these power series both converges in same interval

|x-x0|<R (R>0).

0)()(2

2

yxQdxdyxP

dxyd

)()()(

0

1

xPxPxP

)()()(

0

2

xPxPxQ

0

0 )(m

m xxc

POWER SERIES SOLUTION THEOREM

• c0,c1..are constant and x0 is known as the center of expansion .

0)()()( 212

2

yxPdxdyxP

dxydxPo )...(ieqn

STEPS FOR THE SOLUTION OF O.D.E BY POWER SERIES METHOD

o Find O.P x0 if is not given.o Assume that

o Assuming that term by term differentiation is valid , then differentiate eq. (1) term wise to get y’ , y’’.. And substitute the values in eq.(i).

o Collect the coefficients of like powers of(x-x0) and equate them to “0”, or make the exponent on the x to be the same.

0

0 )(m

mm xxcy )1...(neq

STEPS FOR THE SOLUTION OF O.D.E BY POWER SERIES METHOD

o Collect the coefficients of like powers of(x-x0) and equate them to “0”, or make the exponent on the x to be the same.

o Substituting these values of cm in eq.(1) to get series solution of equation ..(i).

CONCEPT OF FROBENINUS METHOD In above section we have learn that power

series solution of the differential equation about an ordinary point x0.

But when, x0 is regular singular point then an extension of power series method known as “Frobeninus method” or “Generalized power series method”

When x0 is regular singular point then the solution will be

Here, r is unknown constant to be determined.

0

00 )(||m

mm

r xxcxx

SYSTEMATIZED STEPS FOR FROBENINUS METHOD

Consider the differential equation from eq..(i) with a regular singular point x=x0.

Assume that the eq..(i) has a solution of the from

where r, c0, c1,… are constants to be determined, ‘r’ is called “index” and c0, c1, c2,..are coefficients.Here, the eq..(2) is valid in 0<(x-x0)<R.

Assuming that term by term differentiation is valid, we get

0

00 )()(m

mm

r xxcxxy )2...(neq

0

10 )()('

m

rmm xxcrmy

0

20 )()1)((''

m

rmm xxcrmrmy

SYSTEMATIZED STEPS FOR FROBENINUS METHOD On substituting the values of y’, y’’ and y’’’ in

the given eq..(i), we get an algebraic eq with various powers of x.

Equate to zero, theA. Coefficients of the lowest degree terms in x,

assuming c0≠0,this gives a quadratic eq in r, which is known as an “Indicial equation”.

B. Coefficients of general term in x, this gives a relation between the coefficients of two different orders i.e. &

(say). This is called “Recurrence relation”.c. Coefficients of some other powers of x.

2mcmc

SYSTEMATIZED STEPS FOR FROBENINUS METHOD Using the result a & c and employing the

appropriate theorem, the G.S is as

where A and B are arbitrary constants and y1 and y2 are two linearly independent solution.

Further There are FOUR methods to solve the different types of equations.

)(21 )( xByxAyy