1

Answers to Chapter 4

Exercise 4A 1 y � x2 � c where c is constant

x

yy � x2 � 3

y � x2 � 2

y � x2 � 1

y � x2

y � x2 � 1

y � x2 � 2

�2

�1

1

2

3

O

2 y � Aex where A is constant

x

y y � 3ex

y � �3ex

y � 2ex

y � �2ex

y � ex

y � �ex

1

�1

�2

�3

2

3

O

3 y � x3 __

3 � c where c is constant

y � � 2

O x

x3

3

y

�4

�3

�2

�1

1

2

3

4

�2 2

y � x3

3

y � � 1x3

3

y � � 1x3

3

y � � 2x3

3

4 y � ln Ax, where A is constant

5 y � Ax2, where A is constant

O x

yy � 2x2

12

y � x2

y � �2x2

y � �x2

y � x2

12y � � x2

6 y2 � x2 � 2c, where c is constant

O x

y

y2 � x2 � 1

y2 � x2 � �1

y2 � x2 � 16

y2 � x2 � �16

y2 � x2 � 0

y2 � x2 � 4

�4

�3

�2

�1

1

2

3

4

�4 �2�3 �1 2 41 3 y2 � x2 � �4

Answers to Chapter 4

O x

y

�4�3�2�1

1234

�4�6

y � ln 3xy � ln 2xy � ln x

y � ln x

�2 2 4 6

13

y � ln x

y � ln (�3x)y � ln (�2x)

y � ln (�x)

y � ln (� x)y � ln (� x) 1

213

12

2

ANSWERS

7 y � ln 1 ________ (�x � c)

, where c is constant

O x

yy � ln

�4

�3

�2

�1

1

2

3

4

�4�6 �2�3�5 �1 21

12 � x

y � ln 1�x

y � ln 11 � xy � ln 1

(�1 � x)

y � ln( )1�2 � x

8 y � Ax _____ x � 1 , x � 0, where A is constant

O x

y

y �

2

1

3

44x

x � 1

y � 3xx � 1

y � 2xx � 1

y � xx � 1

9 y � sin x � c, where c is constant

O x

y

y � sin x � 1

y � sin x

y � sin x � 1

y � sin x � 2

10 y � A sin x, where A is constant

O x

y

y � 3 sin x

y � �3 sin x

y � �2 sin xy � �sin x

y � 2 sin xy � sin x

�4

�3

�2

�1

1

2

3

4

π2 π

11 x � tan t � c for � � __ 2 � t � � __

2 , where c is constant

O t

x � tan t � 2

x � tan t � 2

x � tan t � 1

x � tan t � 1

x � tan t

x

�4

�6

�2

2

4

6

� π2

π2

12 x � Aet _______

1 � Aet , where A is constant

O t

x

x �

0.5

1

3et

1 � 3et

x � et

1 � et

x � et

1 � et

12

12

13 a

O x

y

y2 � x

y2 � 4x

y2 � 9xy2 � 16x

b y2 � 9x

ANSWERS

3

14 a y2 � 5x2 � 45 b

O x

y

�4

�5

�6

�7

�3

�2

�1

1

2

3

4

5

6

7

�3 3

Exercise 4BIn the following answers, c and A are constants.

1 y � 1 __ x sin x � c __ x

2 y � xe2x � e2x � cex

3 y � 3x cosec x � c cosec x 4 y � xex � cx

5 y � ln [ 1 __ 2 � c __

x2 ]

6 y � � √___________

( 1 __ 6 x2 � c ___

2x ) 7 a y � 1 � 1 __

x � c __

x2

b 1 � 1 __ x � 1 ___

4x2

1 � 1 __ x � 1 __

x2

y � 1 � 1 __ x � 5 __

x2

O x

y

y � 1

y � 1 � �

(�2, )

� 12

1x

14x2

y � 1 � �1x

1x2

34

y � 1 � �1x

5x2

8 a y � ln

A(x � 1) ________

(x � 2) __________

ln x

b y � ln 16 ___

3 (x � 1)

_______ (x � 2)

___________

ln x

Exercise 4CIn the following answers, c is a constant. 1 y � 1 _ 3 e

x � ce�2x

2 y � �cot x � c cosec x 3 y � xecos x � cecos x

4 y � e2x � cex

5 y � ( x2 __

2 � c ) cos x

6 y � 1 __ x ln x � c __ x

7 y � x ln (x � 2) � cx 8 y � 1 _ 4 x � cx � 1 _ 3

9 y � (x � 2) ln (x � 2) � c (x � 2)

10 y � 1 __ x3 e x � 1 __

x4 e x � c __ x4

11 y � 1 __ x e x � 1 __ x2 e x � 1 __

x2

12 y � �1 ___ 3x2 � 4x ___

3

13 a y � 1 _ 3 (x2 � 1)2 � c _______

(x2 � 1)

b y � 1 _ 3 (x2 � 1)2 � 2 _________

3(x2 � 1)

14 a y � 1 � c ____________ sec x � tan x

b y � 1 � 1 ____________ sec x � tan x or y � 1 � cos x _________ 1 � sin x

Exercise 4DIn the following answers, c is a constant and A is a positive constant. 1 y2 � 2x2 (ln x � c) 2 y3 � 3x3 (ln x � c)

3 y � �x _______ ln x � c

4 y3 � x3 (A x � 1)

5 y � √______

cos x ______ 4x � c

6 x � (1 � c e � 1 _ 6 t3

)2

7 y � 4x ____________ 4c � (x � 1)4

8 a y � √______

x � c ______ 1 � x2

b y � √_______

x � 4 ______ 1 � x2

4

ANSWERS

Mixed exercise 4E 1 (x2 � 16 )

1 _ 2 � c, where c is a constant

x

y y � x2 � 16 � 2

y � x2 � 16 � 4

y � x2 � 16

4

6

O

2 a y � e 1 _ 2 x

2

b y � 2e 1 _ 2 x

2

c y � 3e 1 _ 2 x

2

x

y

y � e x2

2

12

1

3

O

y � 2e x212y � 3e x21

2

3 v � 1 __ k [(g � ku)e�kt � g]

t

�gk

v

O

4 y � 2 sin x � c cos x, where c is a constant

5 y � 5 � c (1 � x2 ) 1 _ 2 , where c is a constant

6 y � � x __ 2 � c __ x , where c is a constant

7 y � 2 __ 5 x

3 _ 2 � c __ x , where c is a constant

8 y � 1 _ 2 � c e �x2

, where c is a constant

9 y � 2x � c x √______

1 � x2 , where c is a constant

10 a q � k e �t

__ Rc , where k is a constant

b q � Ec � k e �t

__ Rc , where k is a constant

c q � c ___________ (1 � R2p2c2)

(cos pt � Rpc sin pt) � k' e �t

__ Rc ,

where k' is constant

11 a y � k _____ � a ex � ceax, where c is a constant

b y � (kx � c)eax, where c is a constant

c y � kx n � 1 ______

n � 1 eax � ceax, where c is a constant

12 y � 1 ____________ 1 � cx � ln x , where c is a constant

13 y � √_____

c � x _____ cos x , where c is a constant

14 2y2 (ln y � c) � x2 � 0, where c is a constant

15 y ___

2x � 1 __ 2 ln y � 1 __

2 ln x � c, where c is a constant

16 ln y � 3x2 ___

2y2 � c, where c is a constant

17 y � �1 _____ x � c � x, where c is a constant

18 y � x � 2 � 1 � Ae2x ________

1 � Ae2x , where A is a positive constant