Higher Engineering Mathematics

I

Differential equations

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 anon-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 contain-

ing resistanceR, inductanceLand capacitance

Cin series.

(ii)m

d^2 s

dt^2

+a

ds

dt

+ks=0, defining a mechanical

system, 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.E is

Young’s modulus andIis the second moment

of area.

If D represents

d

dx

and D^2 represents

d^2

dx^2

then the

above 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

Thusy=Aemxis a solution of the given equation pro-

vided 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 factoris-

ing 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).

50.2 Procedure to solve differential

equations of the form

a

d^2 y

dx^2

+b

dy

dx

+cy= 0

(a) Rewrite the differential equation

a

d^2 y

dx^2

+b

dy

dx

+cy= 0

as (aD^2 +bD+c)y = 0

(b) Substitute m for D and solve the auxiliary

equationam^2 +bm+c=0 form.