2011 MSM DIfferential Equation (Solutions)

H2 Math (9740) 2010 MSM: DIFFERENTIAL EQUATION

1. Show, by means of the substitution , that the differential equation

can be reduced to the form

[2]

Hence find y in terms of x, given that when . [4]

[2010/IJC/Prelim/I/6]

[Solution]

Given:

Differentiate w.r.t x

------- (1)

:

(shown)


, where A is a constant

Given that when ,

Thus,


2. (a) Verify that y = x is a particular solution of the differential equation

[2]

(b) Show that the substitution y = ux reduces the differential equation

to the differential equation

[3]

Hence find the general solution of the differential equation

xyyx

xy

2dd 22

[4]

(c)Due to a rapid disease outbreak, the population of fish in a river, x (in thousands), is believed to obey the differential equation

where t is the time in days, and a > 0 is a constant. Given that the entire population of fish is wiped out by the disease eventually, show that the general solution of the differential equation is . [3]

Explain the meaning of a, in the context of the question. Sketch the family of solution curves of the differential equation for a = 1 and 2. [2]

[2010/CJC/Prelim/II/4]

[Solution]

(a) Since y = x and ,

LHS = 1

RHS

x2 x2

2x21=LHS

(b)


(c)

Since entire population is wiped out by the disease eventually, as

Hence, C = 0, D = 0.

a represents the initial population of the fish (in thousands).


3. A particular solution of a differential equation is given by

Show that [2]

A second, related, family of curves is given by the differential equation

By means of the substitution , show that the general solution for y, in

terms of x, is , where c is an arbitrary constant. [3]

Sketch, on a single diagram, three distinct members of the second family of solution curves, stating clearly the coordinates of the points where the curves cross the axes and the equations of any asymptotes.

[5][2010/JJC/Prelim/II/2]

[Solution]

Alternative:


Family of solution curves:

y

1y x

0c

1x 1x

1c

1c 0


4. An innovation is introduced into a community of 100 farmers at time . Let denote the number of farmers who have adopted the innovation at time t.

Assume that is a continuous function of time. The rate at which the number of farmers in that community who adopted the innovation at a particular instant is proportional to the product of the number of farmers who have already adopted and the number of farmers who have not adopted the innovation.

Initially, one farmer adopted the innovation and the rate at which the number of farmers who adopted the innovation is one farmer per unit time.

(i) Show that , where k is to be determined. [1]

(ii) Find the particular solution of , in terms of t. [5]

(iii) Sketch the graph of versus t, for . [2]

(iv) Using the graph in (iii) or otherwise, find the time taken for 75% of the population of farmers to adopt the innovation, leaving your answer to 2 decimal places. [1]

(v) Give a reason why the model may not be suitable. [1][2010/NYJC/Prelim/I/10]

[Solution]

(i)

At ,

therefore

(ii)


ln x −ln 100−x

10099

tC

ln

x100 −x

10099

tC

x100−x

Ae10099

t, where A is a constant.

When , ,

x 100Ae10099

t−Axe

10099

t

x 100Ae

10099

t

1 Ae10099

t

100e10099

t

99 e10099

t

(iii)

(iv) Using GC, 5.64 years.

(v) The farmers may be influenced by adoption of innovation from other sources, e.g. mass media, besides farmers. Or any other reasonable answer.

100

0 t

x


5. In an experiment, Andy and Bob attempt to devise a formula to describe how the volume of water, V m3, in a tank, changes with time at t hours.

(i) Andy gives his formula as Given that when

show that Sketch this solution curve. [4]

(ii) However, Bob believes that it is more likely to be Given that

when , show that the general solution for V can be expressed as

where C is a constant.

Hence, or otherwise, sketch on a single diagram, two distinct members of the family of solution curves. [5]

(iii) It is also given that when in Bob’s model. Suppose that the water

in the tank does not overflow, explain, using your diagrams in parts (i) and (ii), why Andy’s model is more appropriate compared to Bob’s model.

[2][2010/DHS/Prelim/II/3]

[Solution]

(i)

1V

t820ln( )or 2.677


(ii)

(iii) then

Therefore given the above initial condition, Bob’s model corresponds to

solution curve type (I) in part (ii).

Therefore in Bob’s model, the volume of water approaches infinity in the

long run (not realistic) whereas in Andy’s model, the volume of water

reasonably diminishes to zero in the long run/after some time.

Thus, Andy’s model is more appropriate than Bob’s model.

C

tC

V 1(I)4

C

1(II)4

C


6. Newton’s Law of Cooling states that the rate at which the temperature of a body falls is proportional to the amount by which its temperature exceeds that of its surroundings. At time t minutes after cooling commences, the temperature of the body is . Assuming that the room temperature remains constant at and the body has an initial temperature of , show that , where k is an arbitrary constant.

[5]

Given that it takes 8 minutes for the temperature of the body to drop from to , determine how much more time is needed for the body to cool to , leaving your answer to one decimal place.

[3][2010/MJC/Prelim/I/6]

[Solution]

θ −30 Ae−kt (where A is a constant)θ 30 Ae−kt



7. The rate at which a substance evaporates is proportional to the volume of the substance which has not yet evaporated. The initial volume of the substance is A m and the volume which has evaporated at time t minutes is x m 3 . Given that it takes (2ln2) minutes for half of its initial volume to evaporate, show that .Find the additional time needed for three quarters of the substance to evaporate, giving your answer in exact form.

[6][2010/YJC/Prelim/I/11]

[Solution]

, k > 0

ln(A x) = kt + C where C is a constant

ln(A – x) = – kt – C

A – x = e -kt(B) where B = e −C is a constant

x = A – B e-kt


When t = 0, x = 0 B = A

When t = 2 ln 2, x =

(Ans)

When x = ,

t = 4 ln 2

Additional time required = 4 ln2 – 2 ln2

= (2 ln 2) minutes (Ans)


8. An economist is studying how the annual economic growth of 2 countries varies with time. The annual economic growth of a country is measured in percentage and is denoted by G and the time in years after 1980 is denoted by t. Both G and t are taken to be continuous variables.(i) Country A is a developing country and the economist found that G and t are

can be modeled by the differential equation . Given that, when

, , find G in terms of t.[4]

(ii) Comment on the suitability of the above differential equation model to forecast the future economic growth of Country A. [1]

(iii) Country B is a developed country and the economist found that G and t can

be modeled by the differential equation .

Given that Country B has been experiencing decreasing economic growth during the period of study, sketch a member of the family of solution curves of the differential equation model for Country B. Hence, comment on the economic growth of Country B in the long term.

[2][2010/ACJC/Prelim/I/6]

[Solution]

When , ,

Examples of possible comments:The model is not suitable because … The economist is assuming that that there are no fluctuations in the economic

growth in the future. The economist is assuming that the country will enjoy perpetual economic

growth in the long term.


The economist is assuming Country A is always experiencing positive and increasing economic growth in the future.

Factors affecting economic growth remains unchanged.

In the long term, Country B is expected to be still in recession with an economic growth decreasing towards -1%.

t

G

10.51 , 1tG Be B

0


9. An underground storm canal has a fixed capacity of and is able to discharge rainwater at a rate proportional to , the volume of rainwater in the storm canal.On a particular stormy day, rainwater is flowing into the canal at a constant rate of per minute. The storm canal is initially empty. Let be the time in minutes for which the rainwater had been flowing into the storm canal,

(i) show that , where is a positive constant. [4]

A first alarm will be sounded at the control room when the volume of rainwater in the storm canal reaches and a second alarm will be sounded when the storm canal is completely filled. Given that the first alarm was sounded 20 minutes after the rainwater started flowing into the storm canal.

(ii) Find the time interval between the first and second alarm. (Assuming the weather condition remains unchanged). [3]

(iii) Briefly discuss the validity of the model for large values of . [1][2010/AJC/Prelim/II/5]

[Solution]

(i)

When ,

(Shown).

(ii) When , ,

From the GC, 2nd alarm : when

The residents will have 10.7 minutes between the 1st and 2nd alarm.


(iii) which is impossible as the canal has only

a fixed volume of . The model is not valid for large values of t.


10. In a chemical reaction a compound X is formed from a compound Y. The mass in grams of X and Y present at time t seconds after the start of the reaction are x and y respectively. The sum of the two masses is equal to 100 grams throughout the reaction. At any time t, the rate of formation of X is proportional to the mass of Y

at that time. When t = 0, x = 5 and .

(i) Show that x satisfies the differential equation d 0.02(100 )dx xt . [2]

(ii) Solve this differential equation, obtaining an expression for x in terms of t.[4]

(iii) Calculate the time taken for the mass of compound Y to decrease to half its initial value. [2]

(iv) Sketch the solution curve obtained in part (ii) and state what happens to compounds X and Y as t becomes very large. [3]

[2010/SAJC/Prelim/II/2]

[Solution](i) x + y = 100

, where k is a constant/

1.9 = k (100 – 5) k = 0.02

(ii)

When t = 0, x = 5,ln(95) = C

So,


(iii) When t = 0, x = 5 y = 95Initial value of y = 95 Half of initial value = 47.5When y = 47.5, x = 52.5

(iv)

As .

Compound Y will be transformed almost completely to compound X.


11. A long cylindrical metal bar is submerged into iced water. A researcher claims that the rate at which the length, l cm, of the bar is shrinking at any time t seconds is proportional to the volume of the bar at that instant, assuming that the cross-sectional area of the bar remains constant during the shrinking process.

Formulate and integrate a differential equation to show that , where A and k are constants. State the range of values of A and of k. [5]

Given that the initial length of the bar is L, sketch a graph to show the relation between l and t. Comment on the suitability of the claim by the researcher. [3]

It is later found that l and t are related by the equation , where B is some constant. Given that l = 0.5L when t = T, show that the length of the bar

when t = 3T, is given by . [2]

[2010/RVHS/Prelim/II/4]

[Solution]

(note: l > 0)

(shown)

and (Ans)

When t = 0, l = L: A = L

The model suggested by the researcher is not suitable as l 0 when t (i.e. the bar vanished).

l

t

L

0


When t = 3T, = (shown)

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2011 MSM DIfferential Equation (Solutions)