The Chemistry Maths Book, Second Edition

12.2 Homogeneous linear equations 339

so thaty

3

1 = 1 cy

1

is also a solution. The functionsy

1

andy

3

are not regarded as

distinct solutions however because each is merely a multiple of the other; the functions

are said to be linearly dependent. In general, two functions,y

1

andy

2

, are said to be

linearly dependent if there exists a relation

a

1

y

1

(x) 1 + 1 a

2

y

2

(x) 1 = 10 (12.4)

such thata

1

anda

2

are not zero. If (12.4) is true only whena

1

1 = 1 a

2

1 = 10 then the

functions are linearly independent, and neither is a multiple of the other. The

solutionsy

1

andy

2

in Example 12.1 are linearly independent.

We now show that ify

1

(x)andy

2

(x)are two solutions of a linear homogeneous

equation then any linear combination of them,

y 1 = 1 c

1

y

1

1 + 1 c

2

y

2

(12.5)

wherec

1

andc

2

are arbitrary constants, is also a solution. We have

Therefore, substituting into the general homogeneous equation (12.2),

= 10

and the result is zero because both sets of terms in square brackets are zero since

y

1

andy

2

are solutions. This important property of linear homogeneous equations

is called the principle of superposition(it is not true for inhomogeneous equations

or for nonlinear equations). In particular, wheny

1

andy

2

are linearly-independent

solutions, the function (12.5), containing two arbitrary constants, is the general

solutionof the homogeneous equation (see Section 11.2).

+++

























c

dy

dx

px

dy

dx

qxy

2

2

2

2

2

2

() ()

=+ +

























c

dy

dx

px

dy

dx

qxy

1

2

1

2

1

1

() ()

++

















++

(

px c

dy

dx

c

dy

dx

() qx cy cy()

1

1

2

2

11 2 2

))

dy

dx

px

dy

dx

qxy c

dy

dx

c

dy

dx

2

2

1

2

1

2

2

2

2

2

++= +





() ()













dy

dx

c

dy

dx

c

dy

dx

dy

dx

c

dy

dx

c

dy

=+ , = +

1

1

2

2

2

2

1

2

1

2

2

2

22

2

dx