Page 23 - Pra U STPM 2022 Penggal 2 - Mathematics

Mathematics Semester 2 STPM Chapter 4 Differential Equations

Example 5

Determine whether each of the following differential equations is a differential equation with separable variables.
dy
2
(a) = y + 3x y
dx
dy x + y
(b) =
dx x
dy
(c) = e x + y
dx
dy
2
Solution: (a) From = y + 3x y,
dx
dy = y(1 + 3x )
2
dx Rearranging
dy 2 dy
= (1 + 3x ) dx
= u(x) dx
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v(y)
dy
2
Thus, = y + 3x y is a differential equation with separable variables.
dx
dy x + y
(b) = is not a differential equation with separable variables.
dx x
dy
(c) From = e x + y
dx
dy = e e
x y
dx
Penerbitan Pelangi
dy = e dx 4
x
e
dy
Thus, = e x + y is a differential equation with separable variables.
dx

dy
To find the solution of a differential equation with separable variables, integrate the equation = u(x) dx
separately. The result of the integration is v(y)
dy
∫ v(y) + C = u(x) dx + C 2
1 ∫
where C and C are integral constants.
2
1
Rearrange it,
dy
∫
∫
v(y) = u(x) dx + C – C 1
2
∫
= u(x) dx + C C = C – C
2 1
This seems to be the solution of the first order differential equation with separable variables.

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04 STPM Math(T) T2.indd 131 28/01/2022 5:44 PM