The Chemistry Maths Book, Second Edition

316 Chapter 11First-order differential equations

EXAMPLE 11.1The equation

provides a model for a first-order rate process such as radioactive decay or a first-

order chemical reaction, wherex(t)is the amount of reacting substance at time t. It is

shown in Section 11.4 that the general solution of the equation is

x(t) 1 = 1 ae

−kt

where ais the arbitrary constant. In this case, the constant is specified by the amount

of substance at any particular time. Ifx

0

1 = 1 x(0)is the amount at timet 1 = 10 thena 1 = 1 x

0

and

x(t) 1 = 1 x

01

e

−kt

is the particular solution. We note that the application of the initial conditionspecifies

the significance of the arbitrary constant as well as its value, and is a necessary step in

the solution of a physical problem.

In the case of a second-order equation, containing a second derivative, twointegrations

are required to remove the second derivative, so that two arbitrary constants (con-

stants of integration) appear in the general solution. For example, the second-order

equation

(11.5)

is integrated once to give

(11.6)

and a second time to give

y 1 = 1 x

2

1 + 1 ax 1 + 1 b (11.7)

This is the general solution, and two conditions are required to specify the constants.

The general solution of an nth-order differential equation contains narbitrary

constants.*

dy

dx

=+ 2 xa

dy

dx

2

2

= 2

dx

dt

=−kx

*Some differential equations have no general solution, some have one or more particular solutions only, some

have a general solution and some particular solutions not obtained from the general solution. Such cases are not

often met in the physical sciences.