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