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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
n
n
xab )( 0
00
lim 


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
 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
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
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...(n
eq





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, 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 
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!

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Series solution to ordinary differential equations

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