The Chemistry Maths Book, Second Edition

370 Chapter 13Second-order differential equations. Some special functions

For this to be equal to zero for all values of x(within the radius of convergence of the

power series) the coefficient of each power of xmust be zero:

a

1

1 + 1 a

0

1 = 1 0, 2a

2

1 + 1 a

1

1 = 1 0, 3a

3

1 + 1 a

2

1 = 1 0,=

so that

and, in general,a

m

1 = 1 (−1)

m

a

0

2 m!. Therefore,

We recognize the infinite series as the power-series expansion of the functione

−x

:

Thereforey 1 = 1 a

0

e

−x

wherea

0

is an arbitrary constant (see Example 11.4(ii) withk 1 = 1 − 1 ).

0 Exercises 1, 2

EXAMPLE 13.2Use the power-series method to solve the equation

1

By equation (13.2), we have

We can write

=++

=

+

∑

m

m

m

mmax

0

2

21

∞

()()

′′=− =×+×+×

=

−

∑

ymmax a ax

m

m

m

2

2

23

1213243

∞

() ()() ())ax

4

2

+

yaxy max y mm

m

m

m

m

m

m

m

=,′=,′′=−

==

−

=

∑∑ ∑

01

1

2

1

∞∞ ∞

()aax

m

m− 2

dy

dx

y

2

2

+= 0

ex

xx x

m

x

m

m

−

=

=− +

!

−

!

+=

−

!

∑

1

23

23

0



∞

()

yax

ax

m

a

x

m

m

m

m

m

mm

m

m

==

−

!

=

−

!

== =

∑∑ ∑

00

0

0

0

1

∞∞ ∞

()

()

aaa

aa

a

aa

102

10

3

20

22 33

=− , =− =+

!

,=−=−

!

, 

1

Leibniz discovered the differential equation for the sine function in 1693 by a geometric argument. He then

solved the equation by a method equivalent to that given in this example to obtain the power-series representation

of the function. The ability to represent functions as power series was a vital element in the development of the

calculus by both Leibniz and Newton.