The Chemistry Maths Book, Second Edition

12.8 Inhomogeneous linear equations 361

The reduced (homogeneous) equation is

y′′ 1 + 13 y′ 1 + 12 y 1 = 10

and has characteristic equation λ

2

1 + 13 λ 1 + 121 = 1 (λ 1 + 1 1)(λ 1 + 1 2) 1 = 10 , with roots λ

1

1 = 1 − 1

andλ

2

1 = 1 − 2. The general solution of the homogeneous equation, the complementary

function, is therefore

y

h

(x) 1 = 1 c

1

e

−x

1 + 1 c

2

e

− 2 x

and the general solution of the inhomogeneous equation is

0 Exercises 31, 32

The method of undetermined coefficients

This method can be used for many elementary functionsr(x) in (12.75), and is

summarized in Table 12.1 for some of the more important types.

Table 12.1

Term inr(x) Choice ofy

p

1. ce

αx

ke

αx

2. cx

n

(n 1 = 1 0, 1, 2, =) a

0

1 + 1 a

1

x 1 + 1 a

2

x

2

1 +1-1+ 1 a

n

x

n

3. c 1 cos 1 ωx or c 1 sin 1 ωxk 1 cos 1 ωx 1 + 1 l 1 sin 1 ωx


4. ce

αx

1 cos 1 ωx or ce

αx

1 sin 1 ωxe

αx

(k 1 cos 1 ωx 1 + 1 l 1 sin 1 ωx)

The table gives the initial choice of particular integraly

p

corresponding to each function

r(x). The coefficients iny

p

are determined by substitutingy

p

into the inhomogeneous

equation. Example 12.11 demonstrates the method for case 2 in the table. This is

sufficient unless a term iny

p

is already a solution of the corresponding homogeneous

equation. In that case

(a)if the characteristic equation of the homogeneous equation has two distinct roots,

multiplyy

p

by xbefore substitution in (12.75), or

(b)if the characteristic equation has a double root, multiplyy

p

by x

2

before sub-

stitution in (12.75). In addition, ifr(x) is the sum of two or more terms, the total

particular solution is the corresponding sum of the correspondingy

p

’s.

EXAMPLE 12.14Find the general solution of the equationy′′ 1 + 13 y′ 1 + 12 y 1 = 13 e

− 2 x

.

By Example 12.13, the general solution of the homogeneous equation is

y

h

(x) 1 = 1 c

1

e

−x

1 + 1 c

2

e

− 2 x

yx y x y x ce ce x x

hp

xx

() () ()=+ =+ +−+

−−

12

22

7

2

3