where c 0 , c 1 , c 2 , c 3 ,... are constants to be determined. Substitute the representations
(12.157) and (12.158) into the differential equation (12.156) to obtain
c 1 + 2 c 2 x+ 3c 3 x^2 +···+ncnxn−^1 +··· = ln a
[
c 0 +c 1 x+c 2 x^2 +···+cnxn+···
]
(12.159)
In equation (12.159) equate the coefficients of like powers of xand show
c 1 =c 0 ln a
2 c 2 =c 1 ln a
3 c 3 =c 2 ln a
..
.
..
.
(n+ 1)cn+1 =cnln a
..
.
..
.
(12.160)
The general equation
(n+ 1)cn+1 =cnln a or cn+1 =cnln a
(n+ 1)
(12.161)
which holds for n= 0, 1 , 2 , 3 ,... is called a recurrence relation or recurrence formula
associated with the given series and tells one how to select the coefficients in order
to satisfy the differential equation. Recall that c 0 =y(0) = 1 is determined from the
initial value x= 0. Using the recurrence formula (12.161) and the equations (12.160)
one finds
n=0
n=1
n=2
..
.
..
.
n=m
c 1 = ln a
c 2 =
1
2!(ln a)
2
c 3 =^1
3!
(ln a)^3
..
.
..
.
cm=
1
m!(ln a)
m
(12.162)
and consequently the power series expansion for y=axis given by
y=ax= 1 + xln a+x
2
2!
(ln a)^2 +x
3
3!
(ln a)^3 +···+x
m
m!
(ln a)m+··· (12.163)
