Higher Engineering Mathematics

METHODS OF DIFFERENTIATION 297

G

d^2 y

dx^2

=[(− 6 x)(−3e−^3 x)+(e−^3 x)(−6)]

+(−6e−^3 x)

= 18 xe−^3 x−6e−^3 x−6e−^3 x

i.e.

d^2 y

dx^2

= 18 xe−^3 x−12e−^3 x

Substituting values into

d^2 y

dx^2

+ 6

dy

dx

+ 9 ygives:

(18xe−^3 x−12e−^3 x)+6(− 6 xe−^3 x+2e−^3 x)

+9(2xe−^3 x)= 18 xe−^3 x−12e−^3 x− 36 xe−^3 x

+12e−^3 x+ 18 xe−^3 x= 0

Thus wheny= 2 xe−^3 x,

d^2 y

dx^2

+ 6

dy

dx

+ 9 y= 0

Problem 27. Evaluate

d^2 y

dθ^2

whenθ=0given

y=4 sec 2θ.

Sincey=4 sec 2θ,

then

dy

dθ

=(4)(2) sec 2θtan 2θ(from Problem 16)

=8 sec 2θtan 2θ(i.e. a product)

d^2 y

dθ^2

=(8 sec 2θ)(2 sec^22 θ)

+(tan 2θ)[(8)(2) sec 2θtan 2θ]

=16 sec^32 θ+16 sec 2θtan^22 θ

When θ=0,

d^2 y

dθ^2

=16 sec^30 +16 sec 0 tan^20

=16(1)+16(1)(0)= 16.

Now try the following exercise.

Exercise 121 Further problems on succes-

sive differentiation

1. Ify= 3 x^4 + 2 x^3 − 3 x+2 find

(a)

d^2 y

dx^2

(b)

d^3 y

dx^3

[(a) 36x^2 + 12 x (b) 72x+12]

2. (a) Givenf(t)=

2

5

t^2 −

1

t^3

+

3

t

−

√

t+ 1

determinef′′(t)

(b) Evaluatef′′(t) whent= 1

⎡

⎣

(a)

4

5

−

12

t^5

+

6

t^3

+

1

4

√

t^3

(b) − 4. 95

⎤

⎦

In Problems 3 and 4, find the second differ-

ential coefficient with respect to the variable.

3. (a) 3 sin 2[t+cost (b) 2 ln 4θ

(a)−(12 sin 2t+cost) (b)

− 2

θ^2

]

4. (a) 2 cos^2 x (b) (2x−3)^4
   [(a) 4( sin^2 x−cos^2 x) (b) 48(2x−3)^2 ]


5. Evaluatef′′(θ) whenθ=0given
   f(θ)=2 sec 3θ [18]


6. Show that the differential equation
   d^2 y
   dx^2

− 4

dy

dx

+ 4 y=0 is satisfied

wheny=xe^2 x

7. Show that, if P and Q are constants and
   y=P cos(lnt)+Q sin(lnt), then

t^2

d^2 y

dt^2

+t

dy

dt

+y= 0