Page 32 - Pra U STPM 2022 Penggal 2 - Mathematics

Mathematics Semester 2 STPM Chapter 4 Differential Equations
dy
2
2
6. Using the substitution y = vx, find the general solution of the differential equation 2xy = y – 4x .
dx
Show that the particular solution which satisfies the condition y = a (a  0) when x = 1 a is
2
y = 4x(a – x).
2
3
3
7. Show that the differential equation xy 2 dy – x – y = 0 can be transformed into the differential equation
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dv 1 dx
x = by using the substitution y = vx. Hence, find the particular solution of the original equation
dx v 2
given that x = 1, y = 1.
dy
2
3
8. Show that the substitution v = x + y transforms the differential equation (x – 1) – 3x – 3y = 0 into
dv 3v dx
the differential equation = . Hence, find the general solution to the original differential equation.
dx (x – 1)
1 dy
9. By substituting z = , transform the differential equation + y = xy into a differential equation
3
y 2 dx
containing z and x. Solve the equation for z, and, hence, show that the solution of the given differential
1
2

equation for which y = 2 when x = 0 is y = 1 2x + 1 2 — 2 .
dy
10. By substituting u = y , transform the differential equation 2xy = y – 4x into a differential equation
2
2
2
dx
containing u and x. Find the particular solution of the given differential equation for which y = a when
u = 1 a.
2
dy
11. By writing u = 1 , reduce the differential equation dx + x y = y to du – u = –1.
2
x
y
dx
Hence, solve the original differential equation when x = 1, y = 1.
4
4.5 Problems Modelled by Differential
Equations
There are many physical situations in which different variables changes at different rates. Differential equations
always arise when we model these physical situations in mathematics. The following situations show a few
physical problems that involve differential equations.
(a) Assuming a particle falls from rest in a medium that causes the velocity to increase at a rate proportional
to its velocity. By denoting velocity as v and time as t, the rate of increase of velocity can be written
as dv . Hence, the velocity of the particle satisfies the differential equation dv = kv.
dx dt
(b) The rate of decay of a radioactive substance is proportional to the amount of remaining substance.
Hence, the amount of remaining substance, x at any time t can be found by solving the differential equation
dx = –kx.
dt
(c) In a chemical process, a certain substance A continuously changes to another substance B. The rate of such
a change is proportional to the mass of A and inversely proportional to the mass of B, at any time t. The
total mass of A and B at any time remains constant and is equal to N. Hence, the mass of B, n, at time t
can be expressed by the differential equation dn = k (N – n).
dt n
2
(d) The rate of increase of a population due to birth is αx and the rate of decrease due to death is βx where
x is the population number at time t. The population number at any time can be found by solving the
2
differential equation dx = αx – βx .
dt
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04 STPM Math(T) T2.indd 140 28/01/2022 5:44 PM