This document discusses power series solutions and the Frobenius method for solving ordinary differential equations with variable coefficients. It explains that power series can be used to find solutions around ordinary points, while the Frobenius method extends this approach to regular singular points through generalized power series involving an index term. The Frobenius method involves making an ansatz for the solution as a power series with an unknown index, then determining the index and coefficients by substituting into the differential equation and setting terms of different powers of x equal to zero.

1. Advanced Engineering Mathematics
Topic:~ Series solution to ordinary
differential equations
(Power series, Power series solutions, Frobenius method)
MECHANICAL DEPARTMENT

2. Kushal Panchal

3. 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.

4. 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

5.  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
n
n
xab )( 0
00
lim 



6. 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.

7.  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
 yxQ
dx
dy
xP
dx
yd
)(
)(
)(
0
1
xP
xP
xP 
)(
)(
)(
0
2
xP
xP
xQ 




0
0 )(
m
m xxc
0)()()( 212
2
 yxP
dx
dy
xP
dx
yd
xPo )...(ieqn

8. 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...(n
eq

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

10.  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

11. 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...(n
eq





0
1
0 )()('
m
rm
m xxcrmy





0
2
0 )()1)((''
m
rm
m xxcrmrmy

12. 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, the
A. 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
2mc
mc
)(21 )( xByxAyy 

13. 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.

14. Thank You!