The Chemistry Maths Book, Second Edition

11.2 Solution of a differential equation 315

In this chapter we consider only first-order differential equations; second-order

equations are discussed in Chapters 12 to 14. We consider first what is meant by

solvinga differential equation.

11.2 Solution of a differential equation

A differential equation containing an unknown functiony(x)of the variable xhas the

general form

(11.1)

in which some combination (a function) of x, y, and the derivatives of yis equal to

zero. To solve, or integrate, the differential equation is meant to find the function

(or family of functions)y(x)that satisfies the equation. For example, the first-order

equation (see also Section 5.2)

(11.2)

has the solution

y 1 = 1 x

2

1 + 1 c (11.3)

where cis an arbitrary constant, becaused(x

2

1 + 1 c) 2 dx 1 = 12 x. The differential equation

is solved by performing an (indefinite) integration. Thus, integration of both sides of

(11.2) with respect to xgives

(11.4)

In the form (11.3), with the arbitrary constant, the solution is called the general

solution, or complete integral, of the differential equation. We saw in Section 5.2 that

the general solution (11.3) represents a family of curves, one for each value of c,

and that a particular curve of the family is obtained by assigning a particular value

to c. The result is a particular solutionof the differential equation. In the physical

context, the general solution represents a whole family of possible physical situations,

and the choice of particular solution is determined by the nature and state of the

system.

ZZ

dy

dx

dx=;=+ 2 x dx y x c

2

dy

dx

= 2 x

fxy

dy

dx

dy

dx

,, , ,

















=

2

2

... 0