Higher Engineering Mathematics

Ess-For-H8152.tex 19/7/2006 18: 2 Page 715

ESSENTIAL FORMULAE 715

Perpendicular axis theorem:

IfOXandOYlie in the plane of areaAin Fig. FA7,

thenAk^2 OZ=Ak^2 OX+AkOY^2 ork^2 OZ=k^2 OX+k^2 OY

Z

Area A

O

X

Y

Figure FA7

Numerical integration

Trapezoidal rule

∫

ydx≈

(

width of

interval

)[

1

2

(

first+last

ordinates

)

+

(

sum of remaining

ordinates

)]

Mid-ordinate rule

∫

ydx≈

(

width of

interval

)(

sum of

mid-ordinates

)

Simpson’s rule

∫

ydx≈

1

3

(

width of

interval

)[(

first+last

ordinate

)

+ 4

(

sum of even

ordinates

)

+ 2

(

sum of remaining

odd ordinates

)]

Differential Equations

First order differential equations

Separation of variables

If

dy

dx

=f(x) theny=

∫

f(x)dx

If

dy

dx

=f(y) then

∫

dx=

∫

dy

f(y)

If

dy

dx

=f(x)·f(y) then

∫

dy

f(y)

=

∫

f(x)dx

Homogeneous equations

IfP

dy

dx

=Q, wherePandQare functions of both

xand y of the same degree throughout (i.e. a

homogeneous first order differential equation) then:

(i) RearrangeP

dy

dx

=Qinto the form

dy

dx

=

Q

P

(ii) Make the substitution y=vx(where vis a

function ofx), from which, by the product rule,

dy

dx

=v(1)+x

dv

dx

(iii) Substitute for bothyand

dy

dx

in the equation

dy

dx

=

Q

P

(iv) Simplify, by cancelling, and then separate the

variables and solve using the

dy

dx

=f(x)·f(y)

method

(v) Substitutev=

y

x

to solve in terms of the original

variables.

Linear first order

If

dy

dx

+Py=Q, wherePandQare functions of

xonly (i.e. a linear first order differential equation),

then

(i) determine the integrating factor, e

∫

Pdx

(ii) substitute the integrating factor (I.F.) into

the equation

y(I.F.)=

∫

(I.F.)Qdx

(iii) determine the integral

∫

(I.F.)Qdx