The Chemistry Maths Book, Second Edition

344 Chapter 12Second-order differential equations. Constant coefficients

where(a 2 2)

2

1 − 1 b 1 < 10. Let(a 2 2)

2

1 − 1 b 1 = 1 −ω

2

, whereωis real. Then

and the roots are

(12.14)

The particular solutions of the differential equation are therefore

y

1

(x) 1 = 1 e

(−a 22 +iω)x

, y

2

(x) 1 = 1 e

(−a 22 −iω)x

(12.15)

and (whenω 1 ≠ 10 ) the general solution is

y(x) 1 = 1 c

1

y

1

(x) 1 + 1 c

2

y

2

(x)

(12.16)

The trigonometric form of the general solution is (see Example 12.5)

y(x) 1 = 1 e

−ax 22

(d

1

1 cos 1 ωx 1 + 1 d

2

1 sin 1 ωx) (12.17)

whered

1

1 = 1 c

1

1 + 1 c

2

andd

2

1 = 1 i(c

1

1 − 1 c

2

).

12.4 Particular solutions

The two arbitrary constants c

1

and c

2

in the general solution,

y(x) 1 = 1 c

1

y

1

(x) 1 + 1 c

2

y

2

(x)

of a homogeneous second-order differential equation are normally determined in

physical applications by the application of two conditions (two for two constants) in

either of the following two forms.

(a) Initial conditions

The values of the functiony(x)andof its derivativey′(x)are specified for some

value of x:

y(x

0

) 1 = 1 y

0

, y′(x

0

) 1 = 1 y

1

(12.18)

A second-order differential equation with initial conditions is called an initial value

problem. Initial conditions are usually associated with applications in dynamics, when

xis the time variable.

=+

( )

−−

ecece

ax 2 ix ix

12

ωω

λωλ ω

12

22

=− + , =− −

a

i

a

i

2

2

2

1

a

bi











 −=− =− =ωωω