The Chemistry Maths Book, Second Edition

328 Chapter 11First-order differential equations

and the integrated rate equation is

(11.46)

A plot ofln[B] 2 [A]against tis a straight line.

0 Exercises 30–32

11.5 First-order linear equations

A first-order differential equation is called linear if it can be written in the form

(11.47)

The equation is linear in bothdy 2 dxand y. When the right side,r(x), is zero the

equation is called homogeneous; otherwise it is an inhomogeneous equation.

The homogeneous equation

(11.48)

The equation is separable:

and, taking the exponential of each side, the general solution is

(11.49)

where ais the arbitrary constant (if ais chosen as zero theny 1 = 10 is called the trivial

solution).

The inhomogeneous equation

The general linear equation is transformed into a form that can be integrated directly

if (11.47) is multiplied throughout by the function

(11.50)

which, given that , has the differential property

3

(11.51)

dF x

dx

Fxpx

()

= ()()

d

dx

Zpxdx px() = ()

Fx e

px dx

()

()

=

∫

yae

pxdx

=

∫

− ()

dy

y

=−px dx() , lny=−Zpxdx c() +

dy

dx

+=pxy() 0

dy

dx

+=pxy rx() ()

1

00

0

0

[] []

ln

[][]

BA[][]

AB

− BA

=kt

3

This method of solving the general first-order differential equation was discovered by Leibniz in 1694.