Differential Equations

1. A particular species of bear survives only on fresh bamboo shoots. Based on

historical data, it is found that every 1 km2 of bamboo vegetation can feed and support the

habitat of at most 4 bears every year. In a new wildlife park with 225 km2 of bamboo

vegetation, the population growth (due to natural birth and death) of these bears can be

modelled by the differential equation

where is the number of bears, in time years, and is the maximum number of

bears the park can support every year.

(i) Find the value of . [1]

(ii) Given that of the bear population is killed by illegal poachers every year,

modify the given differential equation to include this information. Hence, by

using the substitution , show that the differential equation can be

written as . [4]

(iii) What is the long-term stable population of this particular species of bears in

the park? [6]

(iv) State one assumption about the model. [1]

06/NJC/Promo/Qn12

2. By using the substitution , show that

------------------------ (1)

can be reduced to .

Hence find the general solution for differential equation (1), expressing x in terms of .

Given that , find the particular solution for (1). [6]

07/NJC/MCT/Qn1

3. Use the substitution to find the general solution of the differential equation

. Given that , show that the particular solution is

, where A is a constant to be determined. Prove that in the

general case; , where B and n are integers to be determined.

[8]

07/HCI/MCT/Q4

4. A gardener is weeding at a rate proportional to the amount of weeds present in the garden, and weeds are growing a at a rate inversely proportional to the amount of weeds present. Let x be the amount of weeds present at time t hour. When kg,

the amount of weeds remain constant. Show that , where k is

positive constant.

Initially, the garden has 15 kg of weeds to be cleared, and the gardener takes 2 hours to clear 5 kg of weeds.

(i) Solve the differential equation and find the exact value of k. [5]

(ii) The garden can be rid of weeds by applying weed killer when the garden has 2kg of weeds remaining. Find the time, to the nearest minute, before weed killer is applied. [2]

07/HCI/MCT/Q6

5(a) By using the substitution , show that the differential equation

may be reduced to .

Hence, solve the differential equation, expressing y in terms of x. [5]

(b) A balloon is expanding and at time t seconds, its surface area is s cm2. The expansion is such that the rate of increase of s is inversely proportional to . When the surface area is 900 cm2, it is increasing at a rate of 60cm2s-1.

(i) Show that and solve this differential equation given that

when .[5]

(ii) Find, to the nearest second, the time at which 900cm2. [1]

07/SAJC/MCT/Q7

6. (a) Solve the differential equation , given that

when .

(b) Use the substitution to find the general solution of the differential equation

07/VJC/MCT/Q13

7. Show that the general solution of the differential equation can be

expressed as , where C is an arbitrary constant. Hence, sketch on a single

diagram, solution curves for this equation for C > 0, C = 0, and -1< C < 0, showing clearly in terms of C your axial intercepts and asymptotes where necessary.

[12]

07/HCI/MCT/Q8

Answers:

1. (i) 900 (iii) 687

2. ;

3.

4. (i)

5(a) (b) (i) (ii) 7s

6. (a) (b)