Page 19 - Pra U STPM 2022 Penggal 2 - Mathematics

Mathematics Semester 2 STPM Chapter 4 Differential Equations

4.1 Differential Equations

Introduction
Differential to Differential
Equation
Differential equations INFO VIDEO Equation
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Consider x as an independent variable and y as a dependent variable. An equation that involves at least one
2
n
dy d y d y
derivative of y with respect to x, e.g. , , …, is known as a differential equation.
dx dx 2 dx n
A few examples of differential equations are as follows.
dy
(a) + xy = sin x
2
dx
d y dy
2
(b) 2 – 3 + 4y = x
dx 2 dx
dy
2
(c) (1 + x ) 1 2 2 + 3y = 0
dx
Order of differential equation

The order of a differential equation is the highest order of the derivative found in the differential equation.
In the above examples, equations (a) and (c) are differential equations of the first order because the equations
dy
consist of only derivatives of the first order, . Equation (b) is a differential equation of the second order
dx
2
d y
since it involves a second order derivative, .
dx 2 4
Degree of differential equation
The degree of a differential equation is determined by the power of the highest order of the derivative in the
equation. Hence, equations (a) and (b) are differential equations of the first degree since no derivatives are
dy
raised to any power except one. Equation (c) is of second degree as the highest derivative is of power 2.
dx
In this chapter, only differential equations of the first order and first degree that can be solved by separating the
variables or that can be transformed into such equations will be dealt with in further details.

Example 1

Determine the order and the degree of the following differential equations.
dy
(a) + y = x 2
dx
dy
2
2
(b) x (1 + y) 1 2 2 – (1 + x)y = 0
dx
2
d y dy — 3
= 1 +
(c) 2 1 2 2
dx dx
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