Page 41 - Pra U STPM 2022 Penggal 2 - Mathematics

Mathematics Semester 2 STPM Chapter 4 Differential Equations
(c) Explain what happens as x approaches 100?
(d) Obtain the solution of the differential equation and sketch the solution curve.
(e) Find
(i) the number of turtles after 30 years,
(ii) the time for which the number of turtles is 50.
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14. The rate of destruction of a particular drug is proportional to the amount of drug present in the body.
(a) If x is the amount of drug at time, t, write a differential equation relating the amount of drug and
the time and hence, solve the differential equation.
(b) Initially an amount, Q, of the drug is injected to a body and after a time t = 1 hour the amount of
the drug remaining is 2 Q. Obtain an expression for x in terms of Q and t.
3
(c) The drug is injected again to the body after t = 1 hour, t = 2 hours and t = 3 hours. Find that the
amount of drug remaining in the body immediately after 3 hours.
(d) If the drug is administered at regular intervals of 1 hour for an indefinite period, find the amount
of drug remaining in the body.

Summary

1. A differential equation is an equation involving at least one derivative of y with respect to x,

2
n
dy d y d y 4
e.g. , , …, , where x is the independent variable and y the dependent variable.
dx dx 2 dx n
2. The order of a differential equation is the order of the highest derivative found in the differential equation.

3. The solution of a differential equation that contains of an arbitrary constant is a general solution.

4. The solution of a differential equation that satisfies certain conditions (initial or boundary conditions) is
a particular solution.

dy dy
.
5. If the differential equation = f(x, y) can be expressed as = u(x) v(y), where u(x) and v(y) are
dx dx
functions of x and y, it is a differential equation with separable variables.
dy
6. The linear differential equation + f(x) y = g(x), where f(x) and g(x) are functions of x, can be solved
dx
by multiplying both sides of the equation by an integrating factor e ∫ f(x) dx .

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