Higher Engineering Mathematics

492 DIFFERENTIAL EQUATIONS

For example, if y=4 cos 2x,

then

d^6 y

dx^6

=y(6)=4(2^6 ) cos

(

2 x+

6 π

2

)

=4(2^6 ) cos(2x+ 3 π)

=4(2^6 ) cos(2x+π)

=−256 cos 2x

(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=sinh 2x, then

d^5 y

dx^5

=y(5)

=

25

2

{[1+(−1)^5 ] sinh 2x

+[1−(−1)^5 ] cosh 2x}

=

25

2

{[0] sinh 2x+[2] cosh 2x}

=32 cosh 2x

(vi) Ify=coshax,

y′=asinhax

y′′=a^2 coshax

y′′′=a^3 sinhax, and so on

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

(2 sinh 3x)

=243 sinh 3x

(vii) Ify=lnax,y′=

1

x

,y′′=−

1

x^2

,y′′′=

2

x^3

, and

so 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

Note that ify=lnx,y′=

1

x

; if in equation (7),

n=1 theny′=(−1)^0

(0)!

x^1

(−1)^0 =1 and ify′=

1

x

then(0)!= 1 (Check

that (−1)^0 =1 and (0)!=1 on a calculator).

Now try the following exercise.

Exercise 194 Further problems on higher

order differential coefficients as series

Determine the following derivatives:

1. (a)y(4)wheny=e^2 x(b)y(5)wheny=8e

t

2

[(a) 16 e^2 x(b)

1

4

e

t

(^2) ]

2. (a)y(4)wheny=sin 3t

(b)y(7)wheny=

1

50

sin 5θ

[(a) 81 sin 3t (b)− 1562 .5 cos 5θ]