Chapter 50

Second order differential

equations of the form

a

d

2

y

dx

2

+ b

dy

dx

+cy= 0

50.1 Introduction

An equation of the forma

d^2 y

dx^2

+b

dy

dx

+cy=0, where

a,bandcare constants, is called alinear second
order differential equation with constant coeffi-
cients. When the right-hand side of the differential
equation is zero, it is referred to as ahomogeneous
differential equation. When the right-hand side is not
equal to zero (as in Chapter 51) it is referred to as a
non-homogeneous differential equation.
There are numerous engineering examples of second
order differential equations. Three examples are:

(i) L

d^2 q

dt^2

+R

dq

dt

+

1

C

q=0, representing an equa-

tion for chargeqin an electrical circuit containing

resistanceR, inductanceLand capacitanceCin

series.

(ii) m

d^2 s

dt^2

+a

ds

dt

+ks=0,defining amechanical sys-

tem, wheresis the distance from a fixed point

aftertseconds,mis a mass,athe damping factor

andkthe spring stiffness.

(iii)

d^2 y

dx^2

+

P

EI

y=0, representing an equation for the

deflected profileyof a pin-ended uniform strut

of lengthlsubjected to a loadP.Eis Young’s

modulus andIis the second moment of area.

IfDrepresents

d

dx

and D^2 represents

d^2

dx^2

thentheabove

equation may be stated as

(aD^2 +bD+c)y=0. This equation is said to be in

‘D-operator’form.

Ify=Aemxthen

dy

dx

=Amemxand

d^2 y

dx^2

=Am^2 emx.

Substituting these values intoa

d^2 y

dx^2

+b

dy

dx

+cy= 0

gives:

a(Am^2 emx)+b(Amemx)+c(Aemx)= 0

i.e. Aemx(am^2 +bm+c)= 0

Thus y=Aemx is a solution of the given equation

provided that(am^2 +bm+c)=0.am^2 +bm+c=0is

called theauxiliary equation, and since the equation is

a quadratic,mmay be obtained either by factorizing or

by using the quadratic formula. Since, in the auxiliary

equation,a,bandcare real values, then the equation

may have either

(i) two different real roots (whenb^2 > 4 ac)or

(ii) two equal real roots (whenb^2 = 4 ac)or

(iii) two complex roots (whenb^2 < 4 ac).