The Chemistry Maths Book, Second Edition

12 Second-order differential equations. Constant coefficients

12.1 Concepts

The second-order differential equations that are important in the physical sciences

are linear equations of the general form

(12.1)

When the functionr(x)is zero the equation

(12.2)

is called homogeneous; otherwise it is inhomogeneous.

1

The functionsp(x)andq(x)

are called the coefficientsof the equation. When these coefficients are constants the

solutions of the equation can be expressed in terms of elementary functions, and it

is these linear equations with constant coefficients that are discussed in this chapter.

Examples of such equations are the classical equations of motion for the harmonic

oscillator and for forced oscillations in mechanical and electrical systems, and the

Schrödinger equations for the particle in a box and in a ring. Differential equations

with non-constant coefficients are discussed in Chapter 13.

12.2 Homogeneous linear equations

The general homogeneous second-order linear equation with constant coefficients is

(12.3)

where aand bare constants. It is always possible to find a solution of this equation

that has the forme

λx

whereλis a suitable constant.

dy

dx

a

dy

dx

by

2

2

++= 0

dy

dx

px

dy

dx

qxy

2

2

++=() () 0

dy

dx

px

dy

dx

qxy rx

2

2

++=() () ()

1

Many of the methods and examples found in textbooks can be traced to Euler’s Institutiones calculi

differentialis(1755) and Institutiones calculi integralis(1768 –1770, 3 volumes). Euler was responsible for the

distinction between homogeneous and inhomogeneous equations, between particular and general solutions, the

use of integrating factors, and for the solution of second- and higher-order linear differential equations with

constant coefficients.