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This question paper consists of 3 printed pages and 1 blank page.

OXFORD CAMBRIDGE AND RSA EXAMINATIONS

Advanced Subsidiary General Certificate of EducationAdvanced General Certificate of Education

MEI STRUCTURED MATHEMATICS 4758Differential Equations

Friday 27 JANUARY 2006 Afternoon 1 hour 30 minutes

Additional materials:8 page answer bookletGraph paperMEI Examination Formulae and Tables (MF2)

TIME 1 hour 30 minutes

INSTRUCTIONS TO CANDIDATES

Write your name, centre number and candidate number in the spaces provided on the answerbooklet.

Answer any three questions.

You are permitted to use a graphical calculator in this paper.

Final answers should be given to a degree of accuracy appropriate to the context.

INFORMATION FOR CANDIDATES

The number of marks is given in brackets [ ] at the end of each question or part question.

You are advised that an answer may receive no marks unless you show sufficient detail of theworking to indicate that a correct method is being used.

The total number of marks for this paper is 72.

HN/3 OCR 2006 [R/102/2661] Registered Charity 1066969 [Turn over

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2

1 In an electric circuit, the current, Iamps, at time tseconds is modelled by the differential equation

where k is a positive constant which depends on the capacitor in the circuit.

(i) In the case find the general solution. [8]

(ii) In the case find the solution given that initially the current is 1.5 amps and

State the limiting value of the current as t tends to infinity. [12]

(iii) Show that, for all positive values of k, the complementary function for this differentialequation will tend to zero as t tends to infinity. [4]

2 Three differential equations are to be solved.

(i) By separating the variables, or otherwise, solve equation (1) to find y in terms of x, subject tothe condition when Hence calculate y when giving your answer correctto three significant figures. [8]

(ii) Solve equation (2) to find y in terms of x, subject to the condition when Hencecalculate y when giving your answer correct to three significant figures. [11]

Eulers method is used to solve equation (3). The algorithm is given by

The algorithm starts from with and gives when

(iii) Carry out two more steps of the algorithm to find an approximation for the value of y when

How could you find this value with greater accuracy? [5]x 2.

x 1.8.y 0.034 411h 0.1,x 1, y 0

xr1 xrh,

yr1

yrhy

r .

x 2,x 1.y 0

x 2,x 1.y 0

x2dy

dx3y 1 (1)

x2dy

dx3xy cosx (2)

x2dy

dx3x (y0.1y2) cosx (3)

dI

dt 0.k 9,

k 8,

d2I

dt26

dI

dt kI 6e t,

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4758 January 2006

3

3 A rock of massmkg is dropped from a height of 50 m above the sea. The rock falls under the actionof its weight mgN and a resistance force RN, given by where is the velocityof the rock.

At time tseconds, the rock has fallen a distance xm.

(i) Show that Newtons second law gives the equation

justifying the signs of the terms. [3]

(ii) Solve this differential equation to find v in terms of x. Hence show that the rock hits the water

at a speed of , correct to two decimal places. [7]

When the rock is in the water, the resistance to motion is modelled by Assume that thereis no instantaneous change in the velocity of the rock as it hits the water, and that the only forceson the rock are its weight and the resistance.

(iii) Formulate and solve a differential equation to find a relationship between v and x while therock is under water (you are not required to find v in terms of x). How deep must the waterbe in order for the velocity of the rock to be reduced to [9]

(iv) Use your differential equation from part (iii) to deduce the terminal velocity of the rock underwater. Sketch a graph of v against x for the entire motion of the rock. [5]

4 The following simultaneous differential equations are to be solved.

(i) Show that [5]

(ii) Find the general solution for x in terms of t. [10]

(iii) Hence obtain the corresponding general solution for y. [5]

(iv) Obtain approximate expressions for x and y in terms of t, valid for large t. Hence show that,for large t, x is approximately equal toy. Show that, for small t, this is not necessarily the case.

[4]

d2x

dt22

dx

dt5x 3sin t cos t.

dx

dt x 2y sin t

dy

dt 4x 3y cos t

5 m s1?

R 2mv .

30.54 m s1

vdv

dx g 0.001v2,

vm s1R 0.001mv2,

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This question paper consists of 3 printed pages, 1 blank page and an insert.


Advanced Subsidiary General Certificate of EducationAdvanced General Certificate of Education


Thursday 15 JUNE 2006 Afternoon 1 hour 30 minutes

Additional materials:8 page answer bookletGraph paperMEI Examination Formulae and Tables (MF2)

TIME 1 hour 30 minutes


Write your name, centre number and candidate number in the spaces provided on the answerbooklet.


There is an insert for use in Question 3.


Final answers should be given to a degree of accuracy appropriate to the context. The acceleration due to gravity is denoted by g m s2. Unless otherwise instructed, when a

numerical value is needed, use g = 9.8.





HN/3 OCR 2006 [R/102/2661] Registered Charity 1066969 [Turn over

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2

1 The displacement x at time tof an oscillating system from a fixed point is given by

where

(i) For what value of l is the motion simple harmonic? State the general solution in this case.[3]

(ii) Find the range of values of l for which the system is under-damped. [3]

Consider the case

(iii) Find the general solution of the differential equation. [3]

When , and , where is a positive constant.

(iv) Find the particular solution. [4]

(v) Find the least positive value of t for which [3]

Now consider the case with the same initial conditions.

(vi) Find the particular solution and show that it is never zero for [8]

2 The positive quantities x, y and z are related and vary with time t, where . The value of x isdescribed by the differential equation

When

(i) Solve the equation to find x in terms of t. [9]

The quantity y is related to x by the differential equation . When

(ii) Solve the equation to find y in terms of x. Hence express y in terms of t. [5]

The quantity z is related to x by the differential equation When

(iii) Solve this equation for z in terms of x. Calculate the values of x, y and z when givingyour answers correct to 3 significant figures. [10]

t 1,

t 0, z 3.xdz

dx2z 6x .

t 0, y 4.2xdydx y

t 0, x 1.

dx

dt2x t1.

t 0

t 0.

l 3

x 0.

x0

x.

0x x0

t 0

l 1.

l 0.

x..

2lx.

5x 0,

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4758 June 2006

3

3 Answer parts (i) and (ii) on the insert provided.

Two spherical bodies, Alpha and Beta, each of radius 1000 km, are in deep space. The point A ison the surface of Alpha, and the point B is on the surface of Beta. These points are the closestpoints on the two bodies and the distance AB has the constant value of 8000 km.

A probe is fired from A at a speed of in an attempt to reach B, travelling in a straight line.At time tseconds after firing, the displacement of the probe from A is xkm, and the velocity of theprobe is .

The equation of motion for the probe is

This differential equation is to be investigated first by means of a tangent field, shown on the insert.

(i) Show that the direction indicators are parallel to the v-axis when Show

also that the direction indicators are parallel to the x-axis when Hence

complete the tangent field on the insert, excluding the point [6]

(ii) Sketch the solution curve through and the solution curve through . Hence

state what happens to the probe when the speed of projection is

(A) ,

(B) [6]

(iii) Solve the differential equation to find in terms of x and [6]

(iv) Given that the probe reaches B, state the value of x at which is least. Hence find from yoursolution in part (iii) the range of values of for which the probe reaches B. [6]

4 The simultaneous differential equations

are to be solved for

(i) Show that [6]

(ii) Find the general solution for x in terms of t. Hence obtain the corresponding general solution

for y. [9]

(iii) Given that , when , find the particular solutions for x and y and sketch agraph of each solution. [9]

t 0y 17x 4

d2x

dt22

dx

dt 3x 6.

t 0.

dx

dt 2xy3

dy

dt 5x 4y18

V0

v2

V0.v2

0.05 km s

1

.

0.025 km s1

(0, 0.05)(0, 0.025)

(4000, 0).

x 4000 (v 0) .

v 0 (x 4000) .

vdv

dx

1

(9000x)2

1

(1000x)2.

vkm s1

V0km s1

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Advanced Subsidiary General Certificate of Education

Advanced General Certificate of Education


INSERT

Thursday 15 JUNE 2006 Afternoon 1 hours 30 minutes


This insert should be used in Question 3.

Write your name, centre number and candidate number in the spaces provided at the top of thispage and attach it to your answer booklet.

CandidateCandidate Name Centre Number Number

HN/3 OCR 2006 Registered Charity 1066969 [Turn over

This insert consists of 2 printed pages.

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- 4758 Insert June 2006

Insert for use with Question 3

2000 4000 8000 x0

v

0.02

6000

0.06

0.04

0.02

0

0.04

0.06

2

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Write your name, centre number and candidate number in the spaces provided on the answer booklet.




The acceleration due to gravity is denoted by g m s2. Unless otherwise instructed, when a

numerical value is needed, use g= 9.8.




ADVICE TO CANDIDATES

Read each question carefully and make sure you know what you have to do before starting your

answer.

You are advised that an answer may receive no marks unless you show sufficient detail of the

working to indicate that a correct method is being used.

This document consists of 4 printed pages.

HN/4 OCR 2007 [R/102/2661] OCR is an exempt Charity [Turn over

ADVANCED GCE UNIT 4758/01MATHEMATICS (MEI)

Differential Equations

THURSDAY 25 JANUARY 2007 MorningTime: 1 hour 30 minutes

Additional materials:Answer booklet (8 pages)Graph paperMEI Examination Formulae and Tables (MF2)

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1 The differential equation

is to be solved for .

Consider the case .

(i) Find the general solution. [9]

(ii) Find the particular solution subject to the conditions when andy tends to zero ast tends to infinity. Show that this solution is zero only when and sketch a graph of thesolution. [7]

Consider now the case

(iii) Find the general solution. Find also the particular solution subject to the same conditions as in

part (ii). [You may assume that as ] [8]


(*)

where k is a constant, is to be solved for .

(i) Show that [1]

(ii) In the case , solve the differential equation by separating the variables to find the generalsolution for y in terms of x. [6]

Now assume that .

(iii) Solve the differential equation to show that the general solution is

where A is an arbitrary constant. [7]

(iv) Find the particular solution subject to the condition when . Sketch the graph of

the solution for showing the behaviour of y as x tends to zero. [4]

(v) By using the double angle formulae for and , or otherwise, show that there is a

solution to (*) for which y tends to a finite limit as x tends to zero. State the solution and itslimiting value. [6]

cos 2xsin 2x

0 x 12p,

x 14py 0

y Acosec 2x12kcot 2x

k 0

k 0

d

dx(ln sinx) cotx.

0 x 12p

dy

dx 2ycot 2x k,

t.tet0

k 1.

t 0t 0,y 0

k 2

t 0

d2y

dt2

dy

dt2y ekt

2

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3 In an experiment, a ball-bearing of mass m kg falls through a liquid. The ball-bearing is releasedfrom rest and t seconds later its displacement is x m and its velocity is v m s1. The forces actingon the ball-bearing are its weight and a resistance forceR N. Three models forR are to be considered.

In the first model, where is a positive constant.

(i) Show that Hence show that [7]

(ii) Find an expression for x in terms of t. [4]

In the second model, where is a positive constant.

(iii) Show that and hence find v in terms of x. [7]

In the third model, where is a positive constant. Eulers method is used to solve the

resulting differential equation

The algorithm is given by

(iv) Given and using a step length of 0.1, perform two iterations of the algorithm toestimate v when [5]

The terminal velocity of the ball-bearing is 4 m s1.

(v) Verify the value of given in part (iv). [1]

[Question 4 is printed overleaf.]

k3

t 0.2.k3 1.225

tr1

trh , v

r1 v

r hv

.

r.

dd

v

t g k v= - 3

32 .

k3

R mk v= 332 ,

v

gk2v2

dv

dx 1

k2

R m k2v2,

v g

k1

(1ek1t).dv

dt gk

1v.

k1

R mk1v,

3

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(i) Find the values of x and y for which [3]

(ii) Show that [5]

(iii) Find the general solution for x in terms of t. Hence obtain the corresponding general solutionfor y. [10]

(iv) Given that when find the particular solutions. [3]

(v) Sketch the graph of the solution for x, making clear the behaviour of the curve initially andfor large values of t. [3]

t 0,x y 0

d2x

dt2 4

dx

dt 5x 5.

dx

dt

dy

dt 0.

dd

d

d

x

t x y

y

t

x y

= - - +

= - +

3 10

2 5

,

.

4

OCR 2007 4758/01 Jan 07

Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every

reasonable effort has been made by the publisher (OCR) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the

publisher will be pleased to make amends at the earliest possible opportunity.

OCR is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate

(UCLES), which is itself a department of the University of Cambridge.

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Write your name, centre number and candidate number in the spaces provided on the answer booklet.




The acceleration due to gravity is denoted by g m s2. Unless otherwise instructed, when a

numerical value is needed, use g= 9.8.




There is an insert for use in Question 3.

ADVICE TO CANDIDATES

Read each question carefully and make sure that you know what you have to do before starting

your answer.


This document consists of 4 printed pages and an insert.




MONDAY 18 JUNE 2007 MorningTime: 1 hour 30 minutes

Additional materials:Answer booklet (8 pages)Graph paperMEI Examination Formulae and Tables (MF2)

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1 An object is suspended from one end of a vertical spring in a resistive medium. The other end ofthe spring is made to oscillate and the differential equation describing the motion of the object is

where y is the displacement at time tof the object from its equilibrium position.


(ii) Find the particular solution subject to the conditions when What is theamplitude of the motion for large values of t? [8]

(iii) Find the displacement and velocity of the object when [2]

At the upper end of the spring stops oscillating and the differential equation describingthe motion of the object is now

.

(iv) Write down the general solution. Describe briefly the motion for [3]


where n is a positive constant, is to be solved for

First suppose that

(i) Find the general solution for y in terms of x. [8]

(ii) Use your general solution to find the limit ofy as Show how the value of this limit can

be deduced from the differential equation, provided that tends to a finite limit as

[3]

(iii) Find the particular solution given that when Sketch a graph of the solution in

the case [4]

Now consider the case

(iv) Find y in terms of x, given that y has the same value at as at [9]x 2.x 1

n 2.

n 1.

x 1.y 12

x0.dy

dx

x0.

n 2.

x 0.

xdy

dx2y 1x n,

t 10p.

y..4y

.

29y 0

t 10p,

t 10p.

t 0.y. y 0

y..4y

.

29y 3 cos t,

2

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3 There is an insert for use with part (iii) of this question.

Water is draining from a tank. The depth of water in the tank is initially 1m, and after t minutesthe depth is y m.

The depth is first modelled by the differential equation

where k is a constant.

(i) Find y in terms of tand k. [8]

(ii) If the depth of water is 0.5 m after 1 minute, show that correct to three significantfigures. Hence calculate the depth after 2 minutes. [4]

An alternative model is proposed, giving the differential equation

(*)

The insert shows a tangent field for this differential equation.

(iii) Sketch the solution curve starting at and hence estimate the time for the tank to empty.

[4]

Eulers method is now used. The algorithm is given by where is

given by (*).

(iv) Using a step length of 0.1, verify that this gives an estimate of when forthe solution through and calculate an estimate for y when [6]

(v) Use (*) to show that when the depth of water is less than 1cm the model predicts that is

positive for some values of t. [2]


dy

dt

t 0.2.(0, 1)t 0.1y 0.935 54

y.

tr1 t

rh, y

r1 y

rhy

r,.

(0, 1)

d

d

y

t y t= - +( )0 586 0 1 25. . cos .

k 0.586

d

d

y

t k y t= - +( )1 0 1 25. cos ,

3

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(i) Show that [5]

(ii) Find the general solution for x in terms of t. [8]

(iii) Hence obtain the corresponding general solution for y, simplifying your answer. [4]

(iv) Given that when find the particular solutions. Find the values of and

when Sketch graphs of the solutions. [7]t 0.

dy

dt

dx

dtt 0,x y 0

d2x

dt22

dx

dtx 3e2t.

dd

e

d

d

e

x

t x y

y

t

x y

t

t

= - + +

= - + +

-

-

5 4

9 7 3

2

2

,

.

4

OCR 2007 4758/01 June 07

Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every

reasonable effort has been made by the publisher (OCR) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the

publisher will be pleased to make amends at the earliest possible opportunity.

OCR is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate

(UCLES), which is itself a department of the University of Cambridge.

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This insert should be used in Question 3.

Write your name, centre number and candidate number in the spaces provided and attach the pageto your answer booklet.

This insert consists of 2 printed pages.




INSERT

MONDAY 18 JUNE 2007 MorningTime: 1 hour 30 minutes

CandidateName

CentreNumber

CandidateNumber

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3 (iii)

Spare copy

0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.60

0.2

0.4

0.6

0.8

1.0

y

t

0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.60

0.2

0.4

0.6

0.8

1.0

y

t

2

OCR 2007 4758/01 Insert June 07

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ADVANCED GCE 4758/01MATHEMATICS (MEI)


THURSDAY 24 JANUARY 2008 Morning

Time: 1 hour 30 minutes

Additional materials (enclosed): None

Additional materials (required):

Answer Booklet (8 pages)Graph paperMEI Examination Formulae and Tables (MF2)


Write your name in capital letters, your Centre Number and Candidate Number in the spacesprovided on the Answer Booklet.

Read each question carefully and make sure you know what you have to do before startingyour answer.

Answer anythreequestions.



The acceleration due to gravity is denoted bygm s2. Unless otherwise instructed, when anumerical value is needed, use g= 9.8.



The total number of marks for this paper is72.

You are advised that an answer may receiveno marksunless you show sufficient detail of theworking to indicate that a correct method is being used.


OCR 2008 [R/102/2661] OCR is an exempt Charity [Turn over

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2

1 The differential equation d2y

dt2 + 2

dy

dt + y= f(t) is to be solved for t 0 subject to the conditions that

dy

dt =0 andy = 0 whent= 0.

Firstly consider the case f(t) =2.

(i) Find the solution fory in terms oft. [10]

Now consider the case f(t) =et.

(ii) Explain briefly why a particular integral cannot be of the form aet or atet. Find a particular

integral and hence solve the differential equation, subject to the given conditions. [8]

(iii) Fort> 0, show thaty > 0 and find the maximum value ofy. Hence sketch the solution for t 0.

[You may assume thattket 0 ast for anyk.] [6]

2 A raindrop falls from rest through mist. Its velocity, v m s1 vertically downwards, at time t seconds

after it starts to fall is modelled by the differential equation

(1 + t)dv

dt + 3v= (1 + t)g 3.

(i) Solve the differential equation to show that v = 14g(1 + t) 1 + (1 1

4g)(1 + t)3. [10]

The model is refined and the term3 is replaced by the term2v, giving the differential equation

(1 + t)dv

dt

+ 3v= (1 + t)g 2v.

(ii) Find the solution subject to the same initial conditions as before. [9]

(iii) For each model, describe what happens to the acceleration of the raindrop as t . [5]

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3

3 The population,P, of a species at time tyears is to be modelled by a differential equation. The initial

population is 2000.

At first the model dP

dt =0.5Pis used.

(i) FindP in terms oft. [3]

To take account of observed fluctuations, the model is refined to give dP

dt =0.5P + 170sin 2t.

(ii) State the complementary function for this differential equation. Find a particular integral and

hence state the general solution. [8]

(iii) Find the solution subject to the given initial condition. [2]

The model is further refined to givedP

dt =0.5P + P

2

3 sin2t. This is to be solved using Eulers method.

The algorithm is given by tr+1 = tr+ h, Pr+1 = Pr+ h Pr.

(iv) Using a step length of 0.1 and the given initial conditions, perform two iterations of the algorithm

to estimate the population when t= 0.2. [4]

The population is observed to tend to a non-zero finite limit as t , so a further model is proposed,

given by

dP

dt =0.5P1 P

12 000

1

2

.

(v) Without solving the differential equation,

(A) find the limiting value ofP as t , [3]

(B) find the value ofP for which the rate of population growth is greatest. [4]


dx

dt = 3x+y + 9,

dy

dt = 5x+y + 15,

are to be solved for t 0.

(i) Show that d2x

dt2 + 2

dx

dt + 2x= 6. [5]

(ii) Find the general solution forx. [7]

(iii) Hence find the corresponding general solution fory. [3]

(iv) Find the solutions subject to the conditions that x= y = 0 whent= 0. [4]

(v) Sketch, on separate axes, graphs of the solutions for t 0. [5]

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ADVANCED GCE 4758/01MATHEMATICS (MEI)


THURSDAY 12 JUNE 2008 Morning

Time: 1 hour 30 minutes

Additional materials (enclosed): None

Additional materials (required):

Answer Booklet (8 pages)Graph paperMEI Examination Formulae and Tables (MF2)


Write your name in capital letters, your Centre Number and Candidate Number in the spacesprovided on the Answer Booklet.

Read each question carefully and make sure you know what you have to do before startingyour answer.

Answer anythreequestions.



The acceleration due to gravity is denoted bygm s2. Unless otherwise instructed, when anumerical value is needed, use g= 9.8.



The total number of marks for this paper is72.

You are advised that an answer may receiveno marksunless you show sufficient detail of theworking to indicate that a correct method is being used.


OCR 2008 [R/102/2661] OCR is an exempt Charity [Turn over

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2

1 Fig. 1 shows a particle of mass 2 kg suspended from a light vertical spring. At time t seconds its

displacement isxm below its equilibrium level and its velocity is v m s1 vertically downwards. The

forces on the particle are

its weight, 2g N

the tension in the spring, 8(x+ 0.25g) N

the resistance to motion, 2kv N wherekis apositive constant.

equilibriumlevel

xm

Fig. 1

(i) Use Newtons second law to write down the equation of motion for the particle, justifying the

signs of the terms. Hence show that the displacement is described by the differential equation

d2x

dt2 + k

dx

dt + 4x= 0. [4]

The particle is initially at rest with x= 0.1.

(ii) In the casek= 0, state the general solution of the differential equation. Find the solution, subject

to the given initial conditions. [4]

(iii) In the case k = 2, find the solution of the differential equation, subject to the given initial

conditions. Sketch a graph of the solution for t 0. [11]

(iv) Find the range of values ofkfor which the system is over-damped. Sketch a possible graph of

the solution in such a case. [5]

2 The radioactive substance X decays into the substance Y, which in turn decays into Z. At timethours

the masses, in grams, of X, Y and Z are denoted byx,y and respectively.

Initially there is 8 g of X and there is no Y or Z present.

The differential equation modelling the decay of X is dx

dt = 2x.

(i) Findxin terms oft. [3]

The differential equation modelling the amount of Y is dy

dt =2xy.

(ii) Using your expression forxfound in part(i), solve this equation to findy in terms oft. [9]

(iii) Show thaty > 0 fort> 0. Sketch a graph ofy for t 0. [5]

The differential equation modelling the amount of Z is d

dt =y.

(iv) Without solving this equation, show that x+y + =8. Hence show that =81 et

2

. [5]

(v) Calculate the time required for 99% of the total mass to become substance Z. [2]

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3

3 The differential equationtdy

dt + ky = t, where kis a constant, is to be solved for t1, subject to the

conditiony = 0 whent= 1.

(i) Whenk 1, find the solution for y in terms oftand k. [10]

(ii) Sketch a graph of the solution fork= 2. [2]

(iii) Whenk= 1, find the solution for y in terms oft. [5]

Now consider the differential equationtdy

dt siny= t, subject to the conditiony = 0 whent= 1. This

is to be solved by Eulers method. The algorithm is given bytr+1

= tr+ h, y

r+1= y

r+ h y

r.

(iv) Using a step length of 0.1, perform two iterations of the algorithm to estimate the value ofywhen

t= 1.2. [4]

If the algorithm is carried out with a step length of 0.05, the estimate for y whent= 1.2 isy 0.2138.

(v) Explain with a reason which of these two estimates for y when t = 1.2 is likely to be more

accurate. Hence, or otherwise, explain whether these estimates are likely to be overestimates or

underestimates. [3]


dx

dt =4x 6y 9sin t,

dy

dt =3x 5y 7sin t,

are to be solved.

(i) Show that d2x

dt2 +

dx

dt 2x= 9cos t 3sin t. [6]


(iii) Hence find the corresponding general solution fory. [3]

It is given thatxis bounded ast .

(iv) Show thaty is also bounded as t . [2]

(v) Given also that y = 0 when t = 0, find the particular solutions for x and y. Write down the

expressions forxand y as t . [4]

OCR 2008 4758/01 Jun08

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ADVANCED GCE

MATHEMATICS (MEI) 4758/01Differential Equations

Candidates answer on the Answer Booklet

OCR Supplied Materials:

8 page Answer Booklet Graph paper MEIExamination Formulae and Tables (MF2)

Other Materials Required:

None

Wednesday 21 January 2009

Afternoon

Duration: 1 hour 30 minutes

**447755880011**


Write your name clearly in capital letters, your Centre Number and Candidate Number in the spaces providedon the Answer Booklet.

Use black ink. Pencil may be used for graphs and diagrams only. Read each question carefully and make sure that you know what you have to do before starting your answer. Answer anythreequestions. Donot write in the bar codes. You are permitted to use a graphical calculator in this paper. Final answers should be given to a degree of accuracy appropriate to the context.

The acceleration due to gravity is denoted bygm s2

. Unless otherwise instructed, when a numerical value isneeded, use g= 9.8.


The number of marks is given in brackets[ ]at the end of each question or part question. You are advised that an answer may receive no marks unless you show sufficient detail of the working to

indicate that a correct method is being used. The total number of marks for this paper is72. This document consists of4 pages. Any blank pages are indicated.

OCR 2009 [R/102/2661] OCR is an exempt Charity

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2


d3y

dx3+ 2 d

2y

dx2dy

dx 2y=2

is to be solved.

(i) Write down the auxiliary equation. Show that2 is a root of this equation and find the other tworoots. Hence write down the complementary function. [6](ii) Find the general solution. [3]

Whenx=0, y=0 and whenx=ln2,y=0. Asx ,y tends to a finite limit.(iii) Show thaty= 2e2x+ 3ex 1. [6](iv) Show that y= 0 only when x= 0 or ln 2. Show also that the graph ofy against xhas only one

stationary point, and determine its coordinates. [5]

(v) Sketch the graph of the solution forx0. [4]


dy

dxcosx+y sinx=xcos2x

is to be solved for|x| < 12subject to the condition that y=1 whenx= 0.

(i) Find the solution. [10]

(ii) Sketch the solution curve. [2]

Now consider the differential equation

dy

dxcosx+y sinx=xcosxsinx

for|x| < 12, subject to the condition thaty=1 whenx=0.

(iii) Use Eulers method with a step length of 0.1 to estimate y whenx=0.2. The algorithm is givenbyx

r+1=xr+ h, yr+1=yr+ hy r. [6](iv) Use the integrating factor method and the numerical approximation

0.20

xtanxdx0.002 688to estimatey whenx=0.2. [6]

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3

3 An oil drum of mass 60 kg is dropped from rest from a point A which is at a height of 10 m above a

lake. The oil drum is modelled as a particle that moves vertically. When it is xm below A, its speed

is v m s1. Before it enters the water, the forces acting on it are its weight and a resistance force ofmagnitude 1

4v2 N.

(i) Show that

v

240g v2dv

dx= 1

240

and hence findv2 in terms ofx. [9]

(ii) Show that the speed of the oil drum as it reaches the water is 13.71 m s1, correct to two decimalplaces. [1]

After it enters the water, the forces acting on the oil drum are its weight, a resistance force of magnitude

60v N and a buoyancy force of 90g N vertically upwards.

Assume that the initial speed in the water is 13.71 m s1

and that the oil drum moves vertically.

(iii) Show thattseconds after entering the water its speed is given by v=18.61et 4.9. [8](iv) Calculate the greatest depth below the surface of the water that the oil drum reaches. [6]


dx

dt= 3xy + 7

dy

dt= 2xy + 2are to be solved fort0.

(i) Find the values ofxand y for which dx

dt= dy

dt= 0. [2]

(ii) Show that

d2x

dt2+ 4 dx

dt+ 5x=5. [5]

(iii) Find the general solution for x. [6]

(iv) Find the corresponding general solution for y. [3]

Whent=0, x=4 and y=0.(v) Find the solutions forxand y. [3]

(vi) Sketch the graphs ofxagainst tandy againstt, for t0. Explain how your solution to part(i)relates to your graphs. [5]

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ADVANCED GCE



OCR Supplied Materials:

8 page Answer Booklet Graph paper MEIExamination Formulae and Tables (MF2)


None

Wednesday 20 May 2009

Afternoon


**447755880011**



Use black ink. Pencil may be used for graphs and diagrams only. Read each question carefully and make sure that you know what you have to do before starting your answer. Answer anythreequestions. Donot write in the bar codes. You are permitted to use a graphical calculator in this paper. Final answers should be given to a degree of accuracy appropriate to the context.


. Unless otherwise instructed, when a numerical value isneeded, use g = 9.8.





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2

1 A car travels over a rough surface. The vertical motion of the front suspension is modelled by the

differential equation

d2y

dt2 + 25y =20cos5t,

wherey is the vertical displacement of the top of the suspension and tis time.


Initiallyy = 1 and dy

dt =0.

(ii) Find the solution subject to these conditions. [4]

(iii) Sketch the solution curve fort 0. [4]

A refined model of the motion of the suspension is given by

d2ydt2

+ 2dydt

+ 25y = 20cos 5t.

(iv) Verify thaty = 2sin5tis a particular integral for this differential equation. Hence find the general

solution. [6]

(v) Compare the behaviour of the suspension predicted by the two models. [2]


x

dy

dx + 3y =

sinx

x

is to be solved forx >0.

(i) Find the general solution for y in terms ofx. [9]

Asx 0, y tends to a finite limit.

(ii) Use the approximations sinx x 16x3 and cosx 1 1

2x2 (both valid for small x) to find the

value of the arbitrary constant and the limiting value ofy as x 0. Hence state the particular

solution. [6]

(iii) Show that, wheny = 0, tanx =x. [2]

An alternative method of investigating the behaviour of y for small x is to use the approximation

sinx x 16x3 in the differential equation, giving

xdy

dx + 3y =

x 16x3

x .

(iv) Solve this differential equation and, given that y tends to a finite limit as x 0, show that the

value of the limit is the same as that found in part (ii). [7]

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3

3 (a) An electric circuit has an inductor and a resistor in series with an alternating power source.

The circuit is switched on and after tseconds the current is I amps. The current satisfies the


2dI

dt + 4I =3 cos 2t.

(i) Find the complementary function and a particular integral. Hence state the general solutionforIin terms oft. [8]

Initially the current is zero.

(ii) Find the particular solution. [2]

(iii) Calculate the amplitude of the current for large values oft. Sketch the solution curve for

large values oft. [4]

(b) The displacement,y, of a particle at time tsatisfies the differential equation

dy

dt =2 2y + e

t.

You arenot required to solve this differential equation.

The particle initially has displacement zero. The displacement has only one stationary value,

which is wherey = 9

8. Also the velocity of the particle tends to zero ast .

(i) Without solving the differential equation, use it to find

(A) the gradient of the solution curve whent =0; [2]

(B) the value oftat the stationary value ofy; [3]

(C) the limit ofy as t . [2]

(ii) Hence sketch the solution curve for t 0, illustrating these results. [3]


dx

dt =7x+ 6y + 2e

3t

dy

dt = 12x 10y + 5sin t

are to be solved fort 0.

(i) Show that

d2x

dt2 + 3

dx

dt + 2x= 14e

3t+ 30 sin t. [5]

(ii) Show that this differential equation has a particular integral of the formx= ae3t 9cos t+ 3sin t,

wherea is a constant to be determined.

Hence find the general solution for xin terms oft. [8]

(iii) Find the corresponding general solution for y. [4]

(iv) Show that, for large values oft,x= y when tan t k, wherekis a constant to be determined. [4]

(v) Find the ratio of the amplitudes ofy and xfor large values oft. [3]

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ADVANCED GCE



OCR Supplied Materials: 8 page Answer Booklet Insert for Question 2 (inserted) MEI Examination Formulae andTables (MF2)

Other Materials Required:None


Afternoon


**447755880011**



Use black ink. Pencil may be used for graphs and diagrams only. Read each question carefully and make sure that you know what you have to do before starting your answer. Answer anythreequestions. Donot write in the bar codes. There is aninsertfor use in Question 2. You are permitted to use a graphical calculator in this paper. Final answers should be given to a degree of accuracy appropriate to the context.


. Unless otherwise instructed, when a numerical valueis needed, use g= 9.8.


The number of marks is given in brackets[ ]at the end of each question or part question. You are advised that an answer may receiveno marks unless you show sufficient detail of the working to



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2

1 A particle is attached to a spring and suspended vertically from an oscillating platform. The vertical

displacement,y, of the particle from a fixed point at time tis modelled by the differential equation

d2y

dt2 + 6

dy

dt + 9y= 0.5 sin t.


Initially the displacement and velocity are both zero.

(ii) Find the solution. [5]

(iii) Describe the motion of the particle for large values oft. [2]

(iv) Find approximate values of the velocity and displacement at t= 20. [3]

The motion of the platform is stopped at t= 20and the differential equation modelling the subsequent

motion of the particle isd2y

dt2 + 6

dy

dt + 9y= 0.

(v) Write down the general solution. Sketch the solution curve for t> 20. [5]

2 There is an insert for use with part (b)(i) of this question.

(a) The differential equation

dy

dxy tanx= tanx

is to be solved for|x| < 12.


(ii) Find the equation of the solution curve that passes through the origin and sketch the curve.

[4]

(b) The differential equation

dy

dx y2 tanx= tanx

is to be solved approximately, first by using a tangent field and then by Eulers method.(i) On the insert is a tangent field for the differential equation. Sketch the solution curves

through the origin and through(0, 1). [4]

Eulers method is now used, starting at x= 0, y= 1. The algorithm is given by xr+1 = xr+h,

yr+1 = yr+ hy

r.

(ii) Carry out two steps with a step length of 0.1 to verify that the algorithm gives x= 0.2,y 1.0201. [5]

(iii) Explain why it would be inappropriate to extend this numerical solution as far as x= 1.6.[2]

(iv) How could the accuracy of the estimate found in part(b)(ii)be improved? [1]

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3

3 Fig. 3 shows a small ball projected from a point O over

horizontal ground. The forces acting on the ball are its weight

and air resistance. Its initial horizontal component of velocity

is v1

and its subsequent horizontal velocity xis modelled by

the differential equation

dx

dt = kx,

wherekis a positive constant.x

y

Fig. 3

O

The units of displacement are metres and the units of time are

seconds.

(i) Solve this differential equation to find x in terms of t and hence show that the horizontal

displacement from O is given by x= v

1

k1 ekt. [8]

The balls initial vertical component of velocity isv2

and its subsequent vertical velocity yis modelled

by the differential equation

dy

dt = ky g.

(ii) Solve this differential equation to findy in terms oftand hence show that the vertical displacement

from O is given byy =kv

2+ g

k2 1 ekt g

kt. [10]

(iii) Eliminatetbetween the expressions for xand y to show that y =kv2+ gkv

1

x+ gk2

ln1 kxv

1

.[4]

(iv) In the case v1

=v2

= 10, k= 0.1, determine whether the ball will pass over a 5 m high wall at ahorizontal distance 8 m from O. [2]


dx

dt = 3x 4y + 23,

dy

dt =2x+y 7

are to be solved.

(i) Show that d2x

dt2 + 2

dx

dt + 5x= 5. [5]



Whent= 0, x= 8 andy = 0.

(iv) Find the particular solutions forxand y. [4]

(v) Show that for sufficiently large t,y is always greater than x. [4]

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4

THERE ARE NO QUESTIONS PRINTED ON THIS PAGE.

Copyright Information

OCR is committed to seeking permission to reproduce all third-party content that it uses in its assessment materials. OCR has attempted to identify and contact all copyright holders

whose work is used in this paper. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced in the OCR Copyright

Acknowledgements Booklet. This is produced for each series of examinations, is given to all schools that receive assessment material and is freely available to download from our public

website (www.ocr.org.uk) after the live examination series.

If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible opportunity.

For queries or further information please contact the Copyright Team, First Floor, 9 Hills Road, Cambridge CB2 1GE.

OCR is part of the Cambridge Assessment Group; Cambridge Assessmenti s the brand name of University of Cambridge Local ExaminationsSyndicate (UCLES),which is itself a department

of the University of Cambridge.

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ADVANCED GCE

MATHEMATICS (MEI) 4758/01Differential EquationsINSERT for Question 2


Afternoon


**447755880011**


Write your name clearly in capital letters, your Centre Number and Candidate Number in the boxes above. Use black ink. Pencil may be used for graphs and diagrams only. This insert should be used to answer Question2 part(b)(i). Write your answers to Question 2 part (b)(i) in the spaces provided in this insert, and attach it to your Answer

Booklet.


This document consists of2 pages. Any blank pages are indicated.


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2

2 (b) (i)

x

y

1

1 0

1

2

1










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ADVANCED GCE



OCR Supplied Materials: 8 page Answer Booklet MEI Examination Formulae and Tables (MF2)


Scientific or graphical calculator

Monday 24 May 2010

Afternoon


**447755

88

00

11**



Use black ink. Pencil may be used for graphs and diagrams only. Read each question carefully and make sure that you know what you have to do before starting your answer. Answer anythreequestions. Donot write in the bar codes.

You are permitted to use a graphical calculator in this paper. Final answers should be given to a degree of accuracy appropriate to the context.


. Unless otherwise instructed, when a numerical valueis needed, use g= 9.8.





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2

1 The equation of a curve in thex-yplane satisfies the differential equation

d2y

dx2+ 4 dy

dx+ 8y=32x2.


The curve has a minimum point at the origin.

(ii) Find the equation of the curve. [4]

(iii) Describe how the curve behaves for large negative values ofx. [2]

(iv) Write down an approximate expression for y, valid for large positive values ofx. [1]

(v) Sketch the curve. [3]

(vi) Use the differential equation to show that any stationary point below the x-axis must be aminimum. [4]

2 (a) (i) Find the general solution of

dy

dt+ 2y=e2t. [6]

(ii) Find the solution of

ddt

+ 2 =y,

where y is the general solution found in part (i), subject to the conditions that = 1 andddt

=0 whent=0. [7]


dx

dt+ 2x=sin t

is to be solved.

(i) Find the complementary function and a particular integral. Hence state the general solution.

[6]

(ii) Find the solution that satisfies the condition dx

dt=0 when t=0. [3]

(iii) Find approximate bounds between which xvaries for large positive values oft. [2]

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3

3 Water is leaking from a small hole near the base of a tank. The height of the surface of the water

above the hole isy m at timetminutes.

(i) Consider first a cylindrical tank. The height of the water is modelled by the differential equation

dy

dt= ky,

where kis a positive constant. The height of water is initially 1 m and after 2 minutes it is 0.81 m.

Find y in terms oft, stating the range of values oftfor which the solution is valid. Sketch the

solution curve. [10]

(ii) Now consider water leaking from a conical tank. The height of the water is modelled by the


y2dy

dt= 0.4y.

Find how long it takes the height to decrease from 1 m to 0.81 m. [5]

(iii) Now consider water leaking from a spherical tank. The height of the water is modelled by the


(ay y2)dydt

= 0.4y,

wherea is the diameter of the sphere.

This equation is to be solved by Eulers method. The algorithm is given by tr+1= tr+ h,

yr+1=yr+ hyr. The diameter is 2 m and initially the height is 1 m.

Use a step length of 0.1 to estimate the height after 0.2 minutes. [5]

(iv) For any tank, the velocity of the water leaving the hole is proportional to the square root of the

height of the surface of the water above the hole.

By considering the rate of change of the volume of water, derive the differential equation

dy

dt= ky

for the cylindrical tank in part (i). [4]


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4

4 At timet, the quantities xand y are modelled by the simultaneous differential equations

dx

dt=2x 5y + 9e2t,

dy

dt=x 4y + 3e2t.

(i) Show that d2x

dt2+ 2 dx

dt 3x=3e2t. [5]



Initiallyx=0 andy=2.

(iv) Find the particular solutions. [4]

(v) Describe the behaviour of the solutions as t .

State, with reasons, whether this behaviour is different if the initial value ofy is just less than 2,

and the initial value ofxis still 0. [3]










OCR 2010 4758/01 Jun10

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ADVANCED GCE


Candidates answer on the answer booklet.

OCR supplied materials: 8 page answer booklet

(sent with general stationery) MEI Examination Formulae andTables (MF2)

Other materials required: Scientific or graphical calculator


Afternoon


**447755880011**


Write your name, centre number and candidate number in the spaces provided on theanswer booklet. Please write clearly and in capital letters.

Use black ink. Pencil may be used for graphs and diagrams only. Read each question carefully. Make sure you know what you have to do before starting

your answer.

Answer anythreequestions. Donot write in the bar codes. You are permitted to use a scientific or graphical calculator in this paper. Final answers should be given to a degree of accuracy appropriate to the context.

The acceleration due to gravity is denoted bygm s2. Unless otherwise instructed, whena numerical value is needed, use g= 9.8.


The number of marks is given in brackets[ ]at the end of each question or part question. You are advised that an answer may receiveno marksunless you show sufficient detail

of the working to indicate that a correct method is being used.

The total number of marks for this paper is72. This document consists of4 pages. Any blank pages are indicated.


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2

1 (a) The displacement,xm, of a particle at time tseconds is given by the differential equation

d2x

dt2+ 2 dx

dt+ 5x=4et.


The particle is initially at rest at the origin.



d3y

dx3+ 4 d

2y

dx2+ dy

dx 6y=0

is to be solved.

(i) Show that 1 is a root of the auxiliary equation and find the other two roots. Hence find the

general solution. [5]

Whenx=0, y=1 and dydx

= 4. Asx ,y0.(ii) Find the particular solution subject to these conditions. [4]

(iii) Find the value ofxfor whichy=0. [2]

2 (a) The differential equation

dy

dx+2xy

=ex2 sinx

is to be solved subject to the conditionx=0, y=1.(i) Find the particular solution fory in terms ofx. [9]

(ii) Show that y> 0 for all xand that y has a stationary point when x= 0. State the limitingvalue ofy as|x| . Hence draw a simple sketch graph of the solution, given that thestationary point atx=0 is a maximum. [6]


dy

dx+ 2xy=1

is to be solved numerically subject to the condition x=0, y= 1.(i) Use Eulers method with a step length of 0.1 to estimate y when x=0.2. The algorithm is

given byxr+1=xr+ h,yr+1=yr+ hyr. [4]

(ii) Use the integrating factor method and the approximation0.2

0

ex2

dx0.2027 to estimate ywhenx=0.2. [5]

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3


dy

dx+ ky=cos3x,

wherekis a constant, is to be solved.

(i)Find the complementary function. Hence find the general solution for y in terms ofxandk.

[8]

(ii) Find the particular solution subject to the condition that dy

dx=1 whenx=0. [4]


dy

dx+ ky=2ekx.

(iii) Find the general solution. [6]


d2y

dx2 2 dy

dx=4e2x.

(iv) Using your answer to part (iii), or otherwise, solve this differential equation subject to the

conditions thaty=0 and dydx

=1 whenx=0. [6]

4 The populations of foxes,x, and rabbits, y, on an island at time t are modelled by the simultaneous

differential equations

dxdt= 0.1x+ 0.1y,dy

dt= 0.2x+ 0.3y.

(i) Show that d2x

dt2 0.4 dx

dt+ 0.05x=0. [5]



Initially there arex0

foxes andy0

rabbits.

(iv) Find the particular solutions. [4]

(v) In the casey0= 10x

0, find the time at which the model predicts the rabbits will die out. Determine

whether the model predicts the foxes die out before the rabbits. [7]

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4

THERE ARE NO QUESTIONS PRINTED ON THIS PAGE.




Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download from our public website (www.ocr.org.uk) after the live examination series.





OCR 2011 4758/01 Jan11

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ADVANCED GCE


Candidates answer on the answer booklet.

OCR supplied materials: 8 page answer booklet

(sent with general stationery) MEI Examination Formulae andTables (MF2)

Other materials required: Scientific or graphical calculator

Wednesday 18 May 2011

Morning


**447755880011**


Write your name, centre number and candidate number in the spaces provided on theanswer booklet. Please write clearly and in capital letters.

Use black ink. Pencil may be used for graphs and diagrams only. Read each question carefully. Make sure you know what you have to do before starting

your answer.

Answer anythreequestions. Donot write in the bar codes. You are permitted to use a scientific or graphical calculator in this paper. Final answers should be given to a degree of accuracy appropriate to the context.

The acceleration due to gravity is denoted bygm s2. Unless otherwise instructed, whena numerical value is needed, use g= 9.8.


The number of marks is given in brackets[ ]at the end of each question or part question. You are advised that an answer may receiveno marksunless you show sufficient detail

of the working to indicate that a correct method is being used.

The total number of marks for this paper is72. This document consists of4 pages. Any blank pages are indicated.


RP0I23 Turn over

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2


d2y

dt2+ 4 dy

dt+ 3y=13cos 2t ()

is to be solved.


(ii) Find the particular solution, given that whent=0, y and dydt

are both zero. [6]


d3dt3

+ 4 d2dt2

+ 3 ddt= 26sin2t.

(iii) Show that the general solution may be expressed as =y + c wherey is the general solution of(

) andc is a constant. [2]

(iv) Whent= 0, = 2, ddt= 0 and d2

dt2= 13. Use these conditions to find the particular solution. [7]

2 (a) A curve in thex-yplane satisfies the differential equation

dy

dx 2y

x= x

forx>0.(i) Find the general solution fory in terms ofx. [8]

The curve passes through(1, 0).(ii) Find the equation of this curve. [2]

(iii) Find the coordinates of the stationary point of this curve and find the values to whichy anddy

dx tend as x0. Sketch the curve. [6]


dy

dx= x2 +y2

is to be solved approximately by using a tangent field.

(i) Describe the shape of the isocline for which dy

dx=1. [2]

(ii) Sketch, on the same axes, the isoclines for the cases dy

dx= 1, dy

dx= 2, dy

dx= 3. Use these

isoclines to draw a tangent field. [3]

(iii) Sketch the solution curve through(0, 1). [1](iv) Sketch the solution curve through the origin. [2]

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3

3 (a) A particle of mass 2 kg moves on a horizontal straight line containing the origin O. When its

displacement is xm from O, it is subject to a force of magnitude 2k2xN directed towards O,

wherekis a positive constant.

(i) Show that the velocity,v m s1, of the particle satisfies the differential equation

v

dv

dx= k2

x. [3]

The particle is at rest when x=a, wherea is a positive constant.(ii) Solve the differential equation, subject to this condition. Hence show that, while the particle

moves in the negative direction,

dx

dt= ka2 x2. [6]

Initially the particle is at x=a.(iii) Use the standard integral

1a2 x2 dx=arcsin

x

a + c

to findxin terms oft,kand a. [5]

(b) At time ts, the angle, rad, that a pendulum makes with the vertical satisfies the differential

equation

d

d= 9sin

where = ddt.(i) Solve the differential equation for in terms of subject to the condition = 0 when

= 13. Hence show that, while is decreasing,

d

dt= 32cos 1. [6]

(ii) Starting from = 13whent= 0, use Eulers method with a step length of 0.1 to estimate

whent=0.1. The algorithm is given bytr+1= tr+ h, r+1= r+ hr. State whether this

algorithm can usefully be continued, justifying your answer. [4]


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4 The quantitiesxand y at timetare modelled by the simultaneous differential equations

dx

dt= 3x 2y + 3t,

dy

dt=2x+y + t+ 2.

(i) Show that d2x

dt2+ 2 dx

dt+x= 5t 1. [5]



Whent=0, x=9 andy=0.(iv) Find the particular solutions. [4]

(v) Find approximate expressions for xand y in terms oft, valid for large positive values oft. [3]









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A2 GCE MATHEMATICS (MEI)4758/01 Differential Equations

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2. Unless otherwise instructed, when

a numerical value is needed, use g= 9.8.


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1 Fig. 1 shows a particle of mass 0.5 kg hanging from a light vertical spring. At time tseconds its displacement

isxm below its equilibrium level and its velocity is vm s1vertically downwards. The forces on the particle

are

its weight, 0.5gN,

the tension in the spring, 2.5(x+ 0.2g) N,

the resistance to motion, kvN, where kis a positive constant.

equilibriumlevel

x

m

Fig. 1

(i) Use Newtons second law to write down the equation of motion for the particle, justifying the signs of

the terms. Hence show that the displacement is described by the differential equation

d

2x

dt2 + 2k

dx

dt+ 5x= 0. [4]

The particle is initially at rest withx= 0.1.

(ii) Find the set of values of kfor which the system is

(A) over-damped, (B) under-damped, (C) critically damped.

In each of the cases (A) and (B), sketch a possible displacement-time graph of the motion. [7]

(iii) Sketch a displacement-time graph of the motion of the particle in the case k= 0. [1]

A subsequent motion of the particle is modelled by the differential equation

d

2x

dt2 + 2

dx

dt+ 5x= sin 4t.

(iv) Find the particular solution subject to the conditions that the particle is initially at rest with x= 0. [12]

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2 A population of bacteria grows from an initial size of 1000. After thours the size of the population is P.

After 10 hours the size of the population is 4000.

At first the rate of growth is modelled as being proportional to the size of the population.

(i) Write down a differential equation modelling the population growth and solve it forPin terms of t.

[4]

To allow for constraints on the population growth, the model is revised to give

dP

dt= kP(5000 P),

where kis a constant.

(ii) Solve this differential equation to find tin terms ofP, subject to the given conditions. [9]

(iii) Find the time it takes for the population to reach 4900, giving your answer in hours, correct to two

decimal places. [1]

The model is further refined to give

dP

dt= 10

15P

(5000 P),

where is a constant, and it is observed that the maximum rateof growth occurs whenP= 4000.

(iv) Show that = 4. [5]

Starting from t= 10, P= 4000, Eulers method is used with a step length of 0.2 to solve this differential

equation. The algorithm is given by tr+1

= tr+ h,P

r+1=P

r+ hP

r.

(v) Continue the algorithm for two steps to estimate the size of the population when t= 10.4. [5]

3 Consider the differential equation

dy

dxy =x.

(i) Sketch the isoclinesdy

dx= mfor m= 0, 1, 2. Hence draw a sketch of the tangent field. [3]

(ii) State which of the isoclines is an asymptote to any solution curve. [1]

(iii) Sketch on your tangent field the solution curves through (2, 0) and (0, 2). [3]

(iv) Use the integrating factor method to solve the differential equation foryin terms of x, subject to the

conditiony= 3 whenx= 0. [7]


dy

dxy = sinx.

(v) Find the complementary function and a particular integral. Hence state the general solution. [6]

(vi) Find the solution subject to the conditiony= 3 whenx= 0 and sketch the solution curve. [4]

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department of the University of Cambridge.


dx

dt= x + 2y

dy

dt= x 4y + e

2t

are to be solved.

(i) Eliminateyto obtain a second order differential equation for xin terms of t. Hence find the general

solution forx. [14]

(ii) Find the corresponding general solution fory. [3]

Initiallyx= 5 andy= 0.

(iii) Find the particular solutions. [4]

(iv) Show thaty

x

1

2as t . Show also that there is no value of tfor which

y

x=

1

2. [3]

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1 Some differential equations of the form

d

2y

dx2+ 6

dy

dx+ 9y= f(x)

are to be solved.

First consider the case f(x) =x2

.

(i) Find the general solution foryin terms ofx. [9]

(ii) Find the particular solution subject to the conditionsy= 0,dy

dx= 0 whenx= 0. [5]

Now consider the case f(x) = e3x

.

(iii) Explain why neither ae3x

nor axe3x

will be a particular integral for the differential equation. [1]

(iv) State an appropriate form for a particular integral and hence find the general solution. [7]

(v) State with reasons whether it is possible to have particular solutions for which

(A) yis positive for all values ofx,

(B) yis negative for all values ofx. [2]

2 A parachutist of mass mkg falls vertically from rest. After she has fallenxm, her speed is vm s1. The forces

acting on her are her weight and a resistance force of magnitude mkv2N, where kis a constant.

(i) Show that her motion is modelled by the differential equation

vdvdx

=gkv2

and solve this to show that v2=g

k(1 e

2kx). [8]

(ii) Given that her terminal speed is 55 m s1, calculate k. [1]

When her speed is 54 m s1, she opens her parachute. The motion is now modelled by assuming that the

magnitude of the resistance force instantaneously changes to 0.1mgv N. The time from the parachute

opening is tseconds.

(iii) Formulate and solve a differential equation to find vin terms of t. [8]

(iv) Calculate the time it takes for her speed to reduce to 12 m s1. [1]

(v) Calculate the distance she falls from the point at which she opens her parachute to the point at which

her speed is 12 m s1. [6]

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3 The differential equationxdy

dx2y=x

3sinxis to be solved.

(i) Find the general solution foryin terms ofx. [8]

(ii) Find the particular solution subject to the condition y = 0 when x = . Sketch the solution curve

for 0 x 4. [5]

Now consider the differential equationxdy

dx2y

2= 0.

(iii) Find the general solution foryin terms ofx. [5]

Now consider the differential equationxdy

dx2y

2=x

3sinx.

This is to be solved numerically using Eulers method. The algorithm is given by

xr+1

=xr+ h,y

r+1=y

r+ hy

rwith (x

0,y

0) = (3.14, 0).

(iv) Use a step length of 0.01 to estimateywhenx= 3.16. [5]

(v) How could this estimate be improved? [1]


dx

dt= 2x y + 6,

dy

dt=x 2y + 7,

are to be solved.

(i) Eliminateyto obtain a second order differential equation for xin terms of t. Hence find the general

solution forx. [12]

(ii) Find the corresponding general solution fory. [3]

Initiallyx= 7 andy= 0.

(iii) Find the particular solutions. [4]

As t ,yx k.

(iv) State the value of kand show thaty= kxfor infinitely many values of t. [5]

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4758/01 Jan13 OCR 2013


sinx

y

x

y

x

yy x2 5 6

d

d

d

d

d

d

3

3

2

2

+ - - =

is to be solved.

(i) Show that 2 is a root of the auxiliary equation. Find the other two roots and hence find the generalsolution of the differential equation. [10]

When x 0= , y 1= andx

y0

d

d= . Also, y is bounded as x " 3.


(iii) Write down an approximate solution for large positive values of x . Calculate the amplitude of this

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