Page 35 - Pra U STPM 2022 Penggal 2 - Mathematics

Mathematics Semester 2 STPM Chapter 4 Differential Equations

The differential equation for the motion of the object that involves the
change of velocity v with respect to the displacement x and the velocity v is
dv k
v = – .
dx v 2
Separating the variables and integrating both sides:
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∫
3
∫ v dv = – k dx
v 4 = –kx + C
4

When x = 0, v = u
u 4 u 4
= –k(0) + C ⇒ C =
4 4

Substituting the value of C,
v 4 = –kx + u 4
4 4
4
v = –4kx + u 4
4
4
–4kx + u
\ v = 
Equating the velocity found in (a) and that in (b),
4
3
4
–4kx + u
3
–3kt + u
\  = 
Penerbitan Pelangi Sdn
–4kx + u = (u – 3kt) — 4 3
4
4
\ 4kx = u – (u – 3kt) — 4 3
3
4
Example 16
A research is being done on a particular species of bear on an animal reserve territory. Initially there are 50
bears on the reserve. After t years the number of bears, n, satisfies the differential equation
dn = 1 n(k – n) where k is a constant.
dt 20k
(a) Show that k = 200 if it is known that the rate of growth is 0.9 bear per year when n = 20.
(b) What is the maximum rate of growth?
(c) Explain what happens as n approaches 200?
(d) Obtain the solution of the differential equation and sketch the solution curve.
(e) Find
(i) the number of bears after 77 years,
(ii) the time for which the number of bears is 100.

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