Cambridge International AS and A Level Mathematics Pure Mathematics 1

Integration

P1^

6

The equation d

d

y

x

=   2 x is an example of a differential equation and the process of

solving this equation to find y is called integration.

So the solution of the differential equation d

d

y

x

=   2 x is y = x^2 + c.

Such a solution is often referred to as the general solution of the differential

equation. It may be drawn as a family of curves as in figure 6.1. Each curve

corresponds to a particular value of c.

Particular solutions

Sometimes you are given more information about a problem and this enables

you to find just one solution, called the particular solution.

Suppose that in the previous example, in which

d

d

y

x

=   2 x ⇒ y = x^2 + c

you were also told that when x = 2, y = 1.

Substituting these values in y = x^2 + c gives

    1    =  22   + c

c = − 3

and so the particular solution is

y = x^2 − 3.

This is the red curve shown in figure 6.1.

y

x

c = 2

c = 0

c = –3

O

–3

2

Figure 6.1 y = x^2 + c for different values of c

Recall from Activity 5.4 on page 130

that for each member of a family of

curves, the gradient is the same

for any particular value of x.