H2 Differential Equations AssignmentSolutions

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H2 Differential Equations Tutorial 2010 Meridian Junior College

Page 1 of 4

Assignment: [To be collected in January 2010]

1. (i) Find the general solution of the differential equation

1

3

d

d2

+=

x

x

x

y.

(ii) Find the particular solution of the differential equation for which 2= y when

0= x .

(iii) What can you say about the gradient of every solution curve as ∞±→ x ?

(iv) Sketch, on a single diagram, the graph of the solution found in part (ii), together

with 2 other members of the family of solution curves. (N2008/P1/4)

2. The variables x and y are connected by the differential equationd 1

d 1

y x y

x x y

+ +=

− −.

Show that the substitution u = x + y reduces the equation tod 2

d 1

u

x u=

−, and solve this

differential equation.

Deduce that ( x + y)2 + 2( x – y) = A, where A is an arbitrary constant.

3. TJC/Promo/9

(a) Use the substitution cos y u x= to find the general solution of the differential

equation d tan 1d y y x x

+ = . Hence, find y in terms of x, given that 1 y = when

0 x = . [6]

(b) Patients warded in a particular hospital were infected by an air-borne virus at

a rate of c patients per day. The hospital introduced the use of a vaccination

that can cure those infected at a rate proportional to the number of infected

patients. It was also found that the number of infected patients remains

constant when it reaches 4c.

By denoting x as the number of infected patients after the vaccination was

introduced for t days, find (without solving) a differential equation relating x, t

and c only.

State an assumption that is needed for the differential equation to be valid. [3]




Page 2 of 4

Assignment

1. (i)1

3

d

d2

+=

x

x

x

y

( )

2

2

2

3d1

3 2d

2 1

3ln 1

2

x

y x x

x x

x

x C

= +

=+

= + +

(ii) Given 2= y when 0= x , 2C = .

Therefore particular solution is ( )23ln 1 2

2 y x= + + .

(iii)

2

11

3

d

d

x

x

x

y

+

=

As ∞±→ x , gradient of every solution will tend to zero.

(iii)

2.d 1

d 1

y x y

x x y

+ +=

− −

d d1

d d

11

1

2 (shown)

1

u x y

u y

x x

u

u

u

= +

= +

+= +

−

=−

y

1c = −

O

0c =

2c =




Page 3 of 4

( )

( ) ( )

( ) ( )

( ) ( )

2

2

2

2

d 2

d 1

1 d 2 d

1

22

12

2

2 4 , where 2

2

u

x u

u u x

u u x C

x y x y x C

x y x y x A A C

x y x y A

=−

− =

− = +

+ − + = +

+ − + + = = −

+ + − =

15. TJC/Promo/9

(a) 1tand

d=+ x y

x

y

Givencos

d dsin cos

d d

y u x

y uu x x

x x

=

= − +

Therefore,

1tand

d=+ x y

x

y

becomes

dsin cos cos tan 1

d

uu x x u x x

x− + + =

dcos 1

d1d sec d

ln sec tan

u x

xu x x

u x x c

=

=

= + +

General Solution:

xc x x x y costanseclncos ++=

Given that y = 1 when x = 0,

1 cos 0 ln sec0 tan 0 cos 0

1

c

c

= + +

=

Hence,

( )( )ln sec tan 1 cos y x x x= + +




Page 4 of 4

(b) x is the number of infected patients after the vaccination was introduced for t days.

Therefore the d.e. is defined as

d

d

xc kx

t = − ,

where k is the constant of proportionality.

Given that the number of infected patients remains constant when x = 4c,

d0

d

x

t = when x = 4c.

Therefore

( )0 4

1

4

c k c

k

= −

=

Hence, the d.e. relating x, t and c is

c x

t

x+−=

4

1

d

d.

An assumption needed is that the total number of patients after the vaccination remains

constant throughout. (i.e. no additional patients are introduced to the hospital)

Other Possible Assumptions:

Option 1 (if the patients are only infected in the hospital)

1. All infected patients caught the virus in the hospital, not from any other external

sources.

2.

No patients enter or leave the hospital.3. The virus can only be cured by the vaccination.

Option 2 (if the patients are warded after being infected outside the hospital)

4. All infected patients are given the vaccine.

The virus can be cured only be the vaccination.

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