Download Further Pure 3: Differential Equations

Further Pure 3: Differential Equations
Past Examination Questions
1 (i)
Show that by using the substitution y 
1
, the differential equation
z
dy
 2 y  xy 2
dx
may be written in the form
dz
 2z  x .
dx
(ii)
[3]
Find the general solution of
dz
 2z  x ,
dx
and hence find the general solution of
dy
 2 y  xy 2 .
dx
[5]
OCR P4 May 2005
2.
(i)
(ii)
3. (i)
(ii)
4 (i)
The movement of the needle on an instrument used for measuring the strength of an
electrical current can be modelled by the differential equation
d 2
d
4 2 5
  0 ,
dt
dt
where θ represents the angle turned through from a standard position and t
represents time.
Find the general solution of the differential equation.
[3]

Initially the needle is at rest in a position corresponding to   . Express  in
4
[4]
terms of t.
d
 0 when t = 0].
[Hint: If the needle is initially at rest, this means that
dt
OCR P5 June 2005
Find the general solution of the differential equation
d2y
dy
 6  10 y  10.
[5]
2
dx
dx
Find the particular solution representing a curve which has tangent y = x at the point
(0, 0).
[6]
dy
 1 when x = 0 (since y = x has
[Hint: This means that y = 0 when x = 0 AND
dx
gradient 1).
OCR P5 January 2005
Find the values of the constants λ and μ such that y   x sin 3x   x cos 3x is a
particular integral of the differential equation
d2y
 9 y  sin 3x .
dx 2
[5]
(ii)
Find the particular solution of the differential equation
d2y
 9 y  sin 3x
dx 2
which has a stationary value at the point  16  ,0  .
[6]
OCR P5 June 2004
5 (i)
(ii)
6
Show that an integrating factor of the differential equation
dy
xy

x
dx 1  x 2
[3]
is (1  x 2 ) .
Hence solve the differential equation, given that y = 1 when x = 0, giving your
answer in the form y = f(x).
[5]
OCR P4 May 2004
Find the general solution of the differential equation
dy
 y  e3x ,
dx
giving your answer in the form y = f(x).
[4]
OCR P4 January 2005
7.
It is given that y satisfies the differential equation
d2y
dy
 5  4 y  8 x  10  10cos 2 x .
2
dx
dx
[3]
a) Show that y  2 x  sin 2 x is a particular integral of the given differential
equation.
b) Find the general solution of the differential equation.
[4]
[4]
dy
c) Hence express y in terms of x, given that y = 2 and
= 0 when x = 0.
dx
AQA FP3 June 2006
8.
a) Show that sinx is an integrating factor for the differential equation
dy
 (cot x) y  2 cos x .
[3]
dx

b) Solve this differential equation, given that y = 2 when x  .
[6]
2
AQA FP3 June 2006
9.
[2]
a) Find the roots of the equation m2  2m  2  0 in the form a + ib.
b)
(i) Find the general solution of the differential equation
d2y
dy
 2  2 y  4x .
[6]
2
dx
dx
[4]
dy
(ii) Hence express y in terms of x, given that y = 1 and
= 2 when x = 0.
dx
AQA FP3 January 2006
10.
a) Show that y  x3  x is a particular integral of the differential equation
dy 2 xy
 2
 5x2  1 .
dx x  1
[3]
b) By differentiating  x 2  1 y  c implicitly, where y is a function of x and c is a
c
is a solution of the differential equation
x 1
dy 2 xy
[3]
 2
0.
dx x  1
c) Hence find the general solution of
dy 2 xy

 5x2  1 .
[2]
dx x 2  1
AQA FP3 January 2006
constant, show that y 
11 (i)
(ii)
(iii)
2
1
is an integrating factor for the first-order differential equation
x
dy 1
 y  x ln x .
[3]
dx x
Solve this differential equation, given that y = 1 when x = 1.
[6]
Calculate the value of y when x = 1.2, giving your answer to three decimal places.
[1]
AQA FP3 January 2006
Show that
12.
a) Find the integrating factor for the differential equation
[2]
dy
 y  ex .
dx
b) Find the general solution of this differential equation, giving your answer in the
form
y = f(x).
[4]
AQA P5 June 2005
13.
Find the general solution of the differential equation
d 2 y dy
  2 y  3cos x  4sin x .
dx 2 dx
[11]
AQA P5 June 2005
14.
a) Obtain the roots of the equation
m 2  4m  8  0 ,
giving your answers in the form a  ib .
b) Solve the differential equation
d2y
dy
 4  8 y  8e2 x
2
dx
dx
dy
 2 when x = 0.
given that y = 2 and
dx
[2]
[12]
AQA P5 January 2005
15
1
dy
dz
, where z is a function of x, express
in terms of z and
.
z
dx
dx
b) It is given that y satisfies the differential equation
dy
x2
 yx  y 2 , x  0 .
dx
1
Show that the substitution y  transforms this equation into the differential
z
equation
a) Given that y 
[1]
c)
dz z
1
  2 .
dx x
x
(i) Obtain the general solution of the differential equation
dz z
1
  2 .
dx x
x
[2]
[5]
[3]
(ii) Hence obtain y in terms of x, given that y = 2 when x = ½.
AQA P5 January 2005
16.(a)
(a) Find the general solution of the differential equation
dy
d2 y
+2
+ 2y = 2et.
2
dt
dt
(b) Find the particular solution that satisfies y = 1 and
17.(a)
[6]
dy
= 1 at t = 0.
[6]
dt
EDEXCEL June 2004 P4
(a) Find the general solution of the differential equation
dy
+ 2y = x.
dx
[5]
Given that y = 1 at x = 0,
(b) find the exact values of the coordinates of the minimum point of the particular
[4]
solution curve,
[2]
(c) draw a sketch of this particular solution curve.
EDEXCEL June 2004 P4
18.
dy

 y 1 
dx

3 1
, x > 0.

x  x2
(a) Verify that x3ex is an integrating factor for the differential equation.
[3]
(b) Find the general solution of the differential equation.
[4]
(c) Given that y = 1 at x =1, find y at x = 2.
[3]
EDEXCEL January 2004 P4