Advanced Engineering Mathematics

Topic:~ Series solution to ordinary

differential equations (Power series, Power series solutions, Frobenius method)

MECHANICAL DEPARTMENT

Kushal Panchal

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.

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

......,2

210

0

xbxbbxb n

n

n

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

n

n xab )( 0

0

n

n

nn

xab )( 0

00lim

There is some +ve number R such that the series converges for

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

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

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

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

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).

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

0)()(2

2

yxQdx

dyxP

dx

yd)(

)()(

0

1

xP

xPxP

)(

)()(

0

2

xP

xPxQ

0

0 )(m

m xxc

0)()()( 212

2

yxPdx

dyxP

dx

ydxPo )...(ieqn

I. Find O.P x0 if is not given.II. Assume that

III. 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).

IV. 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

m

m xxcy)1...(neq

v. Substituting these values of cm in eq.(1)

to get series solution of equation ..(i).

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

m

m

r xxcxx

1. Consider the differential equation from eq..(i) with a

regular singular point x=x0.

2. 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.

3. Assuming that term by term differentiation is valid, we

get

0

00 )()(m

m

m

r xxcxxy)2...(neq

0

1

0 )()('m

rm

m xxcrmy

0

2

0 )()1)((''m

rm

m xxcrmrmy

On substituting the values of y’, y’’ and y’’’ in the given eq..(i), we get an algebraic eq with various powers of x.

4. 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.

5. Using the result a & c and employing the appropriate theorem, the G.S is as

2mcmc

)(21 )( xByxAyy

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.

Thank You!