The Chemistry Maths Book, Second Edition

11.5 First-order linear equations 329

Then, multiplying (11.47) byF(x),

(11.52)

and, by virtue of (11.51), the left side of this is the derivative of the productF(x)y:

Equation (11.52) can therefore be written as

and can be integrated to give

(11.53)

This is the general solution of the inhomogeneous equation. The functionF(x)is

called the integrating factorof the differential equation.

EXAMPLES 11.6Linear equations

(i)

In this case,p(x) 1 = 1 − 1 , , and the integrating factor is

(the constant of integration need not be include in the exponent because it

cancels in (11.53)). Then, by formula (11.53),

y 1 = 1 e

x

(3e

x

1 + 1 c) 1 = 13 e

2 x

1 + 1 ce

x

(ii)

The integrating factor is

Fx e e x

xdx x

()

() ln

=

∫

==

2 / 22

dy

dx x

+=yx

2

3

3

e y e e dx c e dx c e c

−−xxx x x

=× +=ZZ33 3+=+

2

Fx e e

px dx x

()

()

=

∫

=

−

Zpxdx x() =−

dy

dx

ye

x

−= 3

2

Fxy Fxrxdx c()=+Z ()()

d

dx

Fxy Fxrx() ()()













=

d

dx

Fxy Fx

dy

dx

dF x

dx

yFx

dy

dx

() () F

()

()













=+=+(()()xpxy

Fx

dy

dx

() +=Fxpxy Fxrx()() ()()