Understanding Engineering Mathematics

x

x

y

y

a = 0

a < 0

Figure 15.2Undamped and damped oscillations.

xincreases. This could represent an oscillating system in a resisting

medium for example.

Exercises on 15.5

1. Solve the following initial value problems:

(i) y′′−y′− 6 y= 0 y( 0 )= 1 y′( 0 )= 0

(ii) 2y′′+y′− 10 y= 0 y( 0 )= 0 y′( 0 )= 1

2. Solve the following boundary value problems:

(i) y′′+ 4 y′+ 13 y= 0 y( 0 )= 0 y

(π

2

)

= 1

(ii) y′′− 4 y′+ 4 y= 0 y( 0 )= 0 y( 1 )= 1

Answers

1. (i)

2

5

e^3 x+

3

5

e−^2 x (ii)

2

9

e^2 x−

2

9

e−^5 x/^2

2. (i) −eπ−^2 xsin 3x (ii) xe^2 (x−^1 )

15.6 The inhomogeneous equation

We now turn to the solution of the full inhomogeneous equation

a

d^2 y

dx^2

+b

dy

dx

+cy=f(x)

We now know the general form of the CF, so we just need to find particular
integrals (PI). These depend on the form of f(x).Infact,inmanycasestheform
of the PI mirrors precisely that of the function f(x)itself. The general approach
is to make a trial solution which has the same form as f(x). This does not