Higher Engineering Mathematics, Sixth Edition

494 Higher Engineering Mathematics

In general, y(n)=ansin

(

ax+

nπ

2

)

(2)

For example, if

y=sin 3x,then

d^5 y

dx^5

=y(^5 )

= 35 sin

(

3 x+

5 π

2

)

= 35 sin

(

3 x+

π

2

)

=243cos3x

(iii) Ify=cosax,

y′=−asinax=acos

(

ax+

π

2

)

y′′=−a^2 cosax=a^2 cos

(

ax+

2 π

2

)

y′′′=a^3 sinax=a^3 cos

(

ax+

3 π

2

)

andsoon.

In general, y(n)=ancos

(

ax+

nπ

2

)

(3)

For example, if y=4cos2x,

then

d^6 y

dx^6

=y(^6 )= 4 ( 26 )cos

(

2 x+

6 π

2

)

= 4 ( 26 )cos( 2 x+ 3 π)

= 4 ( 26 )cos( 2 x+π)

=−256cos2x

(iv) Ify=xa,y′=axa−^1 ,y′′=a(a− 1 )xa−^2 ,

y′′′=a(a− 1 )(a− 2 )xa−^3 ,

andy(n)=a(a− 1 )(a− 2 ).....(a−n+ 1 )xa−n

ory(n)=

a!

(a−n)!

xa−n (4)

whereais a positive integer.

For example, ify= 2 x^6 ,then

d^4 y

dx^4

=y(^4 )

=( 2 )

6!

( 6 − 4 )!

x^6 −^4

=( 2 )

6 × 5 × 4 × 3 × 2 × 1

2 × 1

x^2

= 720 x^2

(v) Ify=sinhax, y′=acoshax

y′′=a^2 sinhax

y′′′=a^3 coshax,and so on

Since sinhaxis not periodic (see graph on page

43), it is more difficult to find a general state-

ment fory(n). However, this is achieved with the

following general series:

y(n)=

an

2

{[1+(−1)n]sinhax

+[1−(−1)n]coshax} (5)

For example, if

y=sinh2x,then

d^5 y

dx^5

=y(^5 )

=

25

2

{[1+(− 1 )^5 ]sinh2x

+[1−(− 1 )^5 ]cosh2x}

=

25

2

{[0]sinh2x+[2]cosh2x}

=32cosh2x

(vi) Ify=coshax,

y′=asinhax

y′′=a^2 coshax

y′′′=a^3 sinhax,andsoon

Since coshaxis not periodic (see graph on page

43), again it is more difficult to find a general

statement fory(n). However, this is achieved with

the following general series:

y(n)=

an

2

{[1−(−1)n]sinhax

+[1+(−1)n]coshax} (6)

For example, ify=

1

9

cosh 3x,

then

d^7 y

dx^7

=y(^7 )=

(

1

9

)

37

2

(2sinh3x)

=243sinh 3x

(vii) Ify=lnax,y′=

1

x

,y′′=−

1

x^2

,y′′′=

2

x^3

,andso

on.

In general, y(n)=(−1)n−^1

(n−1)!

xn

(7)

For example, ify=ln 5x,then

d^6 y

dx^6

=y(^6 )=(− 1 )^6 −^1

(

5!

x^6

)

=−

120

x^6