Download Nightly Homework Questions

Advanced Higher
Mathematics
Nightly Questions
You should attempt a short question
every night during the school week.
Your teacher will tell you what question
to attempt each night.
The questions will usually take you
between 5 and 10 minutes.
Doing a small amount of Advanced Higher
Maths every night will improve your
knowledge and confidence.
Your teacher will go over the answer to
the nightly question in class the next day.
4x  1
in partial fractions.
( x  1)( x  2)
1.
Express
2.
Given f ( x)  x( x  1) 4 , find f (x) in its fully factorised form.
[hint: use the product rule]
3.
Given the curve y 
4.
Express
5.
Differentiate:
x
, find the gradient of the tangent to the curve where x  2 .
x 1
[hint: use the quotient rule]
2
5x  3
in partial fractions. [hint: factorise the denominator first]
x  x  30
2
(a)
f ( x)  e x
2
1
(b)
g ( x)  ln x 2  1
[hint: you may wish to use the laws of logarithms to simplify g (x ) before differentiating]
6.
x2 1
Given f ( x)  2
, find f (x) and simplify your answer.
x 1
7.
x2  6x  3
Express
in partial fractions.
x( x  1) 2
x 2  6x  3 A
B
C
[hint: the denominator contains a repeated linear factor, so let
]
 

2
x x  1 ( x  1) 2
x( x  1)
dy
in its fully factorised form.
dx
8.
Given that y  e 4 x cos 2 x , find
9.
Use the substitution u  x 3  1 to find
10.
Differentiate:
11.
Use the substation u  x 2  4 to find
(a)
x
2
( x 3  1) 5 dx .
f ( x)  sin( e 3x )
x
x 2  4dx .
(b)
g ( x)  e sin 2 x
12.
Given that the quadratic function x 2  3 is irreducible, express
x2  2x  9
in partial
( x  1)( x 2  3)
fractions.
[hint: let
x 2  2x  9
A
Bx  C
]

 2
2
( x  1)( x  3) x  1 x  3
13.
Differentiate f ( x)  x 4 tan 2 x .
14.
Use the substitution u  x 2  1 to evaluate
1
 x( x
2
 1) 9 dx .
0
f ( x)  ln( x 4  1)
(a)
(b)
g ( x)  x ln x , x  0
15.
Differentiate:
16.
Use the substitution u  sin 2x to find  sin 3 2 x cos 2 xdx .
17.
Differentiate f ( x) 
18.
Express
19.
Use the substitution u  x  1 to find
20.
Differentiate:
21.
(a)
Given f ( x)  x 3e x , find f (x) in its fully factorised form.
(b)
tan 1 x
Differentiate f ( x) 
and simplify your answer.
1 x2
sin 2 x
, x  0 , and simplify your answer.
x3
x3
in partial fractions.
x( x  1)( x  1)
 x( x  1)
4
dx .
f ( x)  sin 1 (3x)
(a)
(b)
g ( x)  tan 1 ( x 2 )
2
22.
Use the substitution u  x  1 to find
23.
Express
x
1
in partial fractions.
3x  5 x  2
2
x  1dx .
g ( x)  x sin 1 (2 x)
Differentiate:
25.
Differentiate f ( x)  tan 1 ( x  1) , where x  1, and simplify your answer.
26.
Use
(a)

(a)
f ( x)  cos 1 (3x)
24.
(b)
f ( x)
dx  ln f ( x)  C to find each indefinite integral below.
f ( x)
8x
 x 2  1dx
(b)
sec 2 4 x
 1  tan 4 x dx
27.
y is defined implicitly in terms of x by the equation y 2  xy  4 .
dy
Use implicit differentiation to find
in terms of x and y.
dx
28.
Given f ( x) 
29.
Given y  10 x , use logarithmic differentiation to find
30.
(a)
(b)
31.
(a)
(b)
ln x
, x  0 , find f (x) and f (x) in their simplest form.
x2
dy
in terms of x.
dx
dy d 2 y
d3y
,
and
in fully factorised form.
dx dx 2
dx 3
dny
th
Conjecture an expression for the n derivative
.
dx n
Given y  xe x , find
If x  3sin  , show that
9  x 2  3 cos .
Use the substitution x  3sin  to evaluate
1

0
1
9  x2
dx .
32.
Given f ( x)  ( x  1)( x  2) 3 , find f (x) in its fully factorised form.
33.
Differentiate:
(a)
f ( x)  e tan 4 x
(b)
g ( x)  (2 x  1)e 2 x
34.
x4
1
, show that ln y  4 ln x  ln( e 2 x  1) .
2
1
(a)
Given y 
(b)
Use logarithmic differentiation to find
e2x
dy
in terms of x.
dx
35.
A curve is defined by the parametric equations x  3sin t and y  6 cos t for 0  t  2 .

Find the gradient of the curve at the point where t  .
6
[hint: use parametric differentiation]
36.
(a)
Use the discriminant to show that the quadratic function x 2  x  2 is irreducible.
(b)
Hence express
6x  4
in partial fractions.
( x  3)( x 2  x  2)

2
37.
Use the substitution u  sin x to evaluate  sin 4 x cos xdx .
0
38.
y is defined implicitly in terms of x by the equation y 2  x 2  2 xy .
dy
Use implicit differentiation to find
in terms of x and y.
dx
39.
A curve is defined by the parametric equations x  t 2  1 and y  1  3t 3 for all t.
(b)
dy
in terms of t.
dx
Hence find the equation of the tangent to the curve at the point where t  2 .
(a)
Find
(a)
40.
Use parametric differentiation to find

25  x 2
[hint: use
Express
2
1

dx .
(b)
Evaluate
2
dx .
0
 x
dx  sin 1    C and
a
a x
1
2
1
 4 x
2
5x  4
in partial fractions.
x 2  4x
41.
(a)
42.
Use integration by parts to find
 xe
4x
dx .
a
(b)
2
1
1
 x
dx  tan 1    C ]
2
a
x
a
Hence find 
5x  4
dx .
x 2  4x
(a)
f ( x)  ln(cos 2 x)
43.
Differentiate:
44.
Use integration by parts to evaluate
(b)
g ( x) 
e3x
x3

 x sin xdx .
0
45.
A curve has equation y 3  3xy  3x 2  5 .
(a)
(b)
dy
in terms of x and y.
dx
Find the equation of the tangent to the curve at the point A(2, 1).
Use implicit differentiation to find
46.
A curve is defined by the parametric equations x  t 2  t  1 and y  2t 2  t  2 for all t.
Find the equation of the tangent to the curve at the point where t  1 .
47.
Find the general solution of the differential equation e y
dy
 sec 2 x by separating the
dx
variables.
48.
49.
In a town with a large population, a ‘flu virus is spreading rapidly.
The percentage, P, of the population infected t days after the initial outbreak satisfies the
differential equation
dP
 kP ,
where k is a constant.
dt
(a)
Find the general solution of this differential equation, expressing P in terms of t.
(b)
Given that 0∙5% of the population are initially infected and 2∙5% are infected after
7 days, find the value of k correct to 4 decimal places.
A curve is defined by the parametric equations x  t 2  1 and y  t (t  3) 2 for all t.
(a)
(b)
dy
in terms of t.
dx
Hence find the coordinates of the two stationary points on the curve.
[You do not need to determine the nature of the stationary points.]
Use parametric differentiation to find
50.
11  2 x
in partial fractions.
x2  x  2
(a)
Express
(b)
Hence evaluate
5
11  2 x
dx .
2
 x2
x
3
x
2
51.
Use integration by parts twice to find
52.
Differentiate each function with respect to x:
(a)
53.
f ( x)  3x 2 cos x
g ( x) 
(b)
3x  1
, x 1
x 1
Use an integrating factor to find the general solution of the first order linear differential
equation
dy y
  x,
dx x
54.
sin xdx .
(a)
Given that the quadratic function x 2  5 is irreducible, express
partial fractions.
(b)
x  0.
Hence find
[hint: let
12 x 2  20
in
x( x 2  5)
12 x 2  20 A Bx  C
]
 
x( x 2  5) x x 2  5
12 x 2  20
 x( x 2  5) dx .
5
55.
Use the substitution u  x  4 to evaluate
x
x  4dx .
0
dy
3y

by separating the
dx x  2
variables, expressing the solution in the form y  f (x) .
56.
Find the general solution of the differential equation
57.
(a)
Use the binomial theorem to expand ( x 2  2) 4 .
(b)
Differentiate f ( x)  x 2 tan 1 (2 x) .
58.
Solve the quadratic equation z 2  2 z  17  0 for the complex number z.
[hint: use the quadratic formula]
59.
The complex numbers z and w are given by z  7  6i and w  2  i .
z
Find zw and
in the form a  bi .
w
60.
(a)
Given y  x 3 ln x , find
(b)
d2y
dy
Hence verify that y satisfies the differential equation x
 5x  9 y  0 .
2
dx
dx
61.
dy
d2y
and
.
dx
dx 2
2
Differentiate and simplify where possible:
(a)
f ( x)  e 2 x tan x
(b)
g ( x) 
cos 2 x
, x0
x3
dy
in terms of x.
dx
62.
Given y  x x , where x  0 , use logarithmic differentiation to find
63.
2

Expand and simplify  a   .
a

64.
(a)
Given the complex number z  2  3i , express z and z 2 in the form a  bi ,
where z is the complex conjugate of z.
(b)
The complex number w is given by w 
5
Express w in the form a  bi .
65.
Find the particular solution of the differential equation
x  0 . [hint: separate the variables]
66.
(1  2i) 2
.
7i
dy
 2 xy , given that y  3 when
dx
An arithmetic sequence begins 2, 6, 10, 14, ...
(a)
(b)
Find u 30 , the 30th term of this sequence.
Find S 30 , the sum of the first 30 terms of this sequence.
67.
A curve has equation x 2  4 xy  y 2  11  0 .
dy
Find the value of
at the point (2, 3). [hint: use implicit differentiation]
dx
68.
A curve is defined by the parametric equations x  ln( 1  t 2 ) and y  ln( 1  2t 2 ) for all t.
dy
Use parametric differentiation to find
in terms of t.
dx
69.
A mathematical biologist believes that the differential equation x
dy
 3 y  x 4 models a
dx
process. Use an integrating factor to find the general solution of the differential equation.
[hint: first rewrite the differential equation in the standard form
70.
dy
 P( x) y  Q( x) .]
dx
4x  1
in partial fractions.
2x 2  x  6
(a)
Express
(b)
Hence find
(a)
Write down the common ratio of the geometric sequence beginning
24, 12, 6, 3, ...
(b)
Explain why this sequence has a sum to infinity.
(c)
Find the sum to infinity.
72.
(a)
(b)
Verify that z  1  i is a root of the equation z 3  16 z 2  34 z  36  0 .
Write down a second root of the equation and then find the third root.
73.
(a)
Use the substitution u  2 x (or otherwise) to find
71.
 2x
4x 1
dx .
2
 x6

12
(b)
Evaluate
 sec
0
2
4 xdx .

1
1  4x 2
dx .
74.
Relative to suitable coordinate axes, the position of a golf ball at time t seconds is (x, y),
where x and y are functions of t given by
y  20t  4t 2
x  10t ,
( 0  t  5 ),
and x and y are measured in metres.
dy
in terms of t.
dx
(a)
Use parametric differentiation to obtain an expression for
(b)
The speed, v metres per second, of the golf ball at time t seconds is given by
2
2
 dx   dy 
v      .
 dt   dt 
Use this formula to find the speed of the golf ball after 2 seconds.
75.
The first term of an arithmetic sequence is 2 and the 20th term is 97.
(a)
(b)
Find the common difference.
Find the sum of the first 50 terms of this sequence.
(a)
Use the standard formula for
n
76.
 r to find an expression for
r 1
n
 (3r  2) in terms
r 1
of n.
16
(b)
Hence, or otherwise, evaluate
 (3r  2) .
r 1
77.
Find the general solution of the second order differential equation
d2y
dy
 6  9 y  e2 x .
2
dx
dx
78.
(a)
Write down and simplify the general term in the binomial expansion of
( x 2  3 x) 8 .
(b)
Hence, or otherwise, obtain the coefficient of x 13 in the expansion.
n
[hint: the general term in the binomial expansion of (a  b) n is  a n  r b r ]
r
x
.
x 1
[You do not need to determine the nature of the stationary points.]
79.
Find the coordinates of the stationary points on the graph of y 
80.
The equation of a curve is y 
(a)
(b)
81.
x 2  4x  5
, x  1.
x 1
Write down the equation of the vertical asymptote.
Find the equation of the non-vertical asymptote. [hint: use long division first]
Use the substitution u  x  2 to find
[hint: remember that
a
2
x
2
1
dx .
 4x  8
1
1
 x
dx  tan 1    C ]
2
a
x
a
82.
The equation x 2 y  xy 3  5 defines y implicitly as a function of x.
dy
Use implicit differentiation to find
in terms of x and y.
dx
83.
Use integration by parts to evaluate:
1
(a)
2
3x
 xe dx
2
(b)
x
3
ln xdx
1
0
84.
(a)
(b)
(c)
Find the Maclaurin series for f ( x)  e 2 x up to and including the term in x 4 .
Find the Maclaurin series for g ( x)  cos 3x up to and including the term in x 4 .
Hence find the Maclaurin series for h( x)  e 2 x cos 3x up to and including the
term in x 3 .
85.
(a)
Find the number of terms, n, in the arithmetic series 8 + 11 + 14 + ... + 56.
[hint: form the equation u n  56 ]
(b)
Hence find the sum of the series.
(a)
2 

Expand and simplify  x 3  2  .
x 

(b)
Differentiate: (i)
4
86.
f ( x)  ln( 1  e 2x )
(ii)
g ( x) 
sin x
, x0
x
x3
, x  2.
x2
[You do not need to determine the nature of the stationary points.]
87.
Find the coordinates of the stationary points on the graph of y 
98.
Differentiate f ( x)  e tan x cos 2 x and simplify your answer.
89.
(a)
Given the complex number z  1 3i , write down z and express (z ) 2 in the
form a  bi , where z is the complex conjugate of z.
(b)
90.
Find
91.
(a)
(b)
92.
93.
94.
The function f is defined on the set of all real numbers by f ( x)  x 3 sin x .
By considering f ( x) , determine whether the function f is odd, even or neither.
3x  5
 ( x  1)( x  2)( x  3) dx .
[hint: use partial fractions]
x 3 e sin x
1
, show that ln y  3 ln x  sin x  ln( x 2  1) .
2
x 1
dy
Use logarithmic differentiation to find
in terms of x.
dx
Given y 
2
(a)
The function f is defined by f ( x)  3 cos 2 x , 0  x   .
Sketch the graph of y  f (x) .
(b)
On a separate diagram, sketch the graph of y  f (x) .
(a)
Write down the derivative of cos 1 x .
(b)
By using a substitution, or otherwise, find
(c)
Use integration by parts to find  cos 1 xdx .

x
1 x2
dx .
Use an integrating factor to find the general solution of the first order linear differential
equation below.
dy
 3 y  8e  x
dx
95.
96.
(a)
Given xy  x  4 , use implicit differentiation to find
(b)
d2y
Hence find
in terms of x and y.
dx 2
A curve is defined by the parametric equations x 
(a)
Use parametric differentiation to find
(b)
Hence find
dy
in terms of x and y.
dx
1 2
1
t  2t and y  t 3  3t for all t.
2
3
dy
in terms of t.
dx
d2y
in terms of t.
dx 2
x2  6
in partial fractions.
x3  2 x
97.
Express
98.
Use integration by parts twice to find
99.
(a)
[Hint: start by factorising the denominator]
x
2
e 2 x dx .
Write down and simplify the general term in the binomial expansion of
10
 2 2
x   .
x

(b)
Hence, or otherwise, obtain the coefficient of x 8 in the expansion.
n
[hint: the general term in the binomial expansion of (a  b) n is  a n  r b r ]
r
100.
A solid is formed by rotating the area under the line y  3 x between x  1 and x  2
through 360 about the x-axis.
Calculate the volume of the solid of revolution that is formed.
101.
Use Gaussian elimination to solve the following system of equations.
x + y + 3z = 2
2x + y + z = 2
3x + 2y + 5z = 5
102.
The equation of a curve is y 
(a)
(b)
(c)
103.
x 2  3x  6
, x  2 .
x2
Write down the equation of the vertical asymptote.
Find the equation of the non-vertical asymptote. [hint: use long division first]
Find the coordinates of the stationary points on the curve and justify their nature.
The velocity, v metres per second, of a particle moving in a straight line, t seconds after
passing through a fixed point O is
v = 6t2  5t + 3,
104.
105.
t  0.
(a)
(b)
Find the acceleration of the particle when t = 3.
Find the displacement of the particle from O when t = 4.
(a)
Given the complex number z  1  3i , find the modulus and argument of z and
hence write z in polar form.
(b)
The function f is defined on the set of all real numbers by f ( x)  x 2 cos x .
By considering f ( x) , determine whether the function f is odd, even or neither.
1
in partial fractions.
y (1  y )
(a)
Express
(b)
Show that the general solution of the differential equation
dy
 y (1  y )
dx
can be expressed in the form
ln y  ln( 1  y )  x  C
for some constant C.
(c)
106.
1
ex
Given that y  when x  0 , find the value of C and hence show that y 
.
2
1 ex
A solid is formed by rotating the area under the curve y  4  x 2 between x  0 and
x  2 through 360 about the x-axis.
Calculate the volume of the solid of revolution that is formed.
4
107.
(a)
2

Expand and simplify  3 x   .
x

(b)
Use Gaussian elimination to solve the following system of equations.
x+y=4
x + 2y − z = 5
3y − 2z = 6
108.
 1 2
 .
Let A be the matrix A  
 2 4
(a)
(b)
Show that A2  pA for some constant p.
Hence, or otherwise, determine the value of q such that A4  qA .
109.
(a)
(b)
Write the complex number z  3  i in polar form.
Use de Moivre’s theorem to find z 6 in the form a  bi .
110.
A car begins from rest and travels along a straight road.
The velocity, v metres per second, of the car is given by v 
100t
, where t is the time
2t  11
in seconds.
Find the acceleration of the car after 4 seconds.
111.
Consider the system of equations
x+y+z=2
4x + 3y  z = 4
5x + 6y + 8z = 11.
112.
(a)
Use Gaussian elimination to express z in terms of  and hence state the value of
 for which the system has no solution.
(b)
Solve the system of equations when   2 .
 4 p
 q  6
 and B  
 .
Matrices A and B are defined by A  
 2 1 
1 3 
(a)
(b)
Write down the matrix A , where A denotes the transpose of A.
Find the values of p and q if B  3 A .
113.
114.
(a)
Write the complex number 2  2 3i in polar form.
(b)
Hence find the two complex numbers z for which z 2  2  2 3i , expressing each
root in the form r (cos   i sin  ) .
Let I   e 2 x cos xdx .
1
By integrating by parts twice, show that I  e 2 x (sin x  2 cos x)  C .
5
115.
The area bounded by the curve with equation y  1  sin 2 x between x  0 and x 
is rotated through 360 about the x-axis.
Find the volume of the solid of revolution that is formed.
116.
A particle moves along a straight line.
The velocity, v metres per second, of the particle after t seconds is given by
v  t 3  12t 2  32t .
(a)
Find an expression for the acceleration of the particle after t seconds.
(b)
At time t  0 , the particle is at the origin O.
(i)
(ii)
117.
Find an expression for the displacement of the particle from O after t
seconds.
The particle returns to O after T seconds. Find T.
2 

 .
Let A be the 2  2 matrix A  
 2   3
(a)
Find the values of  such that the matrix is singular.
[Hint: remember that A is singular when det A  0 ]
(b)
Write down the inverse matrix matrix A 1 when   3 .

2
118.
(a)
Solve the following system of equations using Gaussian elimination.
2x + y + 3z = 5
x
+ 2z = 3
4x + 3y + z = 3
119.
x3
 1  x 8 dx .
(b)
Use the substitution u  x 4 to find
(a)
Given u = –2i + 5k, v = 3i + 2j – k and w = –i + j + 4k, calculate u . (v × w).
(b)
Find the point of intersection of the line
x  3 y  2 z 1


4
1
2
and the plane with equation 2 x  y  z  4 .
[Hint: express the equation of the line in parametric form first]
120.
The complex number z is such that z  2  z  i , where z  x  yi .
Show that ax  by  c  0 for suitable values of a, b and c.
121.
When a valve is opened, the rate at which water drains from a pool is proportional to the
square root of the depth of the water.
dh
h

This can be represented by the differential equation
, where h is the depth (in
dt
10
metres) of the water and t is the time (in minutes) elapsed since the valve was opened.
122.
h C
t
for some constant C. [hint: separate the variables]
20
(a)
Show that
(b)
Given that the water was initially 9 metres deep when the valve was opened, find
the value of C and hence find the time taken to drain the pool.
Obtain the equation of the plane passing through the points A(2, 1, 1), B(1, 2, 3) and
C(3, 0, 1).
[Hint: find the vectors AB and AC , then the normal vector is n  AB  AC ]
123.
Find the values of m for which the matrix
m 1 1 


A   0 m  2
1 0 1 


is singular.
[Hint: remember that A is singular when det A  0 ]
124.
(a)
Write down and simplify the general term in the binomial expansion of
9
1 

 2x  2  .
x 

(b)
Hence, or otherwise, obtain the term independent of x in the expansion.
n
[hint: the general term in the binomial expansion of (a  b) n is  a n  r b r ]
r
125.
Consider the system of equations
4x + 6z = 1
2x − 2y + 4z = 1
x + y + z = 2
Use Gaussian elimination to express z in terms of  and hence state the value of  for
which the system has no solution.
126.
The lines L1 and L2 are given by the equations
x 1 y
z 3


2
1
1
and
x4 y3 z3


,
1
1
2
respectively.
127.
(a)
Show that L1 and L2 intersect and find the point of intersection.
[Hint: first express the equations of both lines in parametric form]
(b)
Calculate the acute angle between the lines.
(a)
Use the Euclidean algorithm to obtain the greatest common divisor of 1139 and
629.
(b)
Express the greatest common divisor in the form 1139 p  629q for integers p
and q.
128.
(a)
Use a direct proof to prove that the product of an odd integer and an even integer
is always even.
[Hint: Let a be an odd integer, so that a  2k  1 for some integer k.
Let b be an even integer, so that b  2m for some integer m.]
(b)
Prove by contradiction that if n 2 is even, then n is also even, where n is a natural
number.
[Hint: Let n 2 be even and assume that n is odd.]
n
129.
Prove by induction that for all positive integers n,
 4r
3
 n 2 (n  1) 2 .
r 1
Verify that z  1 2i is a root of the equation z 3  3z 2  5 z  25  0 .
Hence find all the roots of this equation.
130.
(a)
(b)
131.
An arithmetic sequence has first term a and common difference d.
The third term, u3 , is equal to 11 and the tenth term, u10 , is equal to 32.
Find the values of a and d.
132.
Three vectors u, v and w are given by
u = 5i + 13j, v = 2i + j + 3k, w = i + 4j – k.
Calculate u . (v × w) and interpret your answer geometrically.
133.
  1  3


The 3 3 matrix A is given by A   0 1
2 .
1 2 1 


(a)
Obtain an expression for det A in terms of  .
(b)
Hence find the value of  for which the matrix A is singular.
n
134.
Prove by induction that for all positive integers n,
1
 r (r  3)  3 n(n  1)(n  5) .
r 1
135.
An arithmetic sequence begins 5, 7, 9, 11, ...
(a)
(b)
Find a formula for S n , the sum of the first n terms of this sequence.
Hence find the value of n for which S n  320 .
136.
Plane  1 has equation 2 x  3 y  z  5 and plane  2 has equation x  y  z  0 .
Calculate the size of the acute angle between the planes 1 and  2 .
[Hint: the angle between two planes is the angle between their normal vectors]
137.
Prove by induction that for all positive integers n, 5 n  3 is divisible by 4.
138.
Show that the line of intersection of the planes x  y  z  6 and 2 x  3 y  2 z  2 has
parametric equations
x,
y  4  14 ,
z  5  20 .
[Hint: let x   in the equation of each plane and solve the system of equations for
y and z]
139.
140.
 4 
 .
Let A be the 2  2 matrix A  
 2 1
(a)
Find A2 .
(b)
Find the value of  for which A2 is singular.
[Hint: remember that A 2 is singular when det( A 2 )  0 ]
(a)
Write down the 2  2 matrix M1 associated with an anti-clockwise rotation of
radians about the origin.
(b)
Write down the matrix M 2 associated with reflection in the y-axis.
(c)
Find the 2  2 matrix, M 3 , associated with an anti-clockwise rotation of
radians about the origin followed by reflection in the y-axis.
141.
(d)
What single transformation is associated with the matrix M 3 ?
(a)
Find the general solution of the second order differential equation
d2y
dy
 5  6 y  12 x 2  2 x  5 .
2
dx
dx
(b)
dy
 3.
dx
Find the particular solution of the differential equation.
When x  0 , y  6 and

2

2