The Chemistry Maths Book, Second Edition

130 Chapter 5Integration

is the operator form of equation (5.2). The operatorsDandD

− 1

have the property

D

− 1

D 1 = 1 D D

− 1

1 = 11

true of any pair of inverse operators. Then

is equivalent to

Value of the integration constant

In an actual problem the value of the integration constantCin (5.2) is determined by

some auxiliary condition. Consider, for example, a curve whose gradient at every

point is given by the function 2 x. The equation of such a curve is

(5.5)

This equation represents a family of curves, one curve for each value of C(Figure 5.3).

If it is required that the curve pass through a particularpoint, the point (1, 1 2)say, the

auxiliary condition on (5.5) is thaty 1 = 12 whenx 1 = 11 ; that is,y 1 = 121 = 111 + 1 C. Therefore,

C 1 = 11 , andy 1 = 1 x

2

1 + 11 is the equation of the particular curve whose gradient is given by

dy 2 dx 1 = 12 xand that passes through the point (1, 1 2).

It is shown in later chapters how, in a physical problem, the value of the integration

constant is determined by the nature and state of the system.

yxdxxC==+Z 2

2

ZZ

d

dx

Fx C dx

d

dx

()+ Fx C dx Fx() ()













=+













=+CC

DD DD

−−













=+













=+

11

Fx C() Fx C Fx C() ()

2

−10+1

(1,2)

C=− 1 , 0 , 1

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Figure 5.3