Page 26 - Pra U STPM 2022 Penggal 2 - Mathematics

Mathematics Semester 2 STPM Chapter 4 Differential Equations

4.3 First Order Linear Differential Equations

dy
The first order linear differential equation is of the form + f(x)y = g(x) where f(x) and g(x) are functions of
dx
x. It can be solved by multiplying both sides of the equation with e ∫ f(x) dx which is called the integrating factor;
Penerbitan Pelangi Sdn Bhd. All Rights Reserved.
dy
+ f(x)y = g(x)
dx
4
3 dy + f(x)y e ∫ f(x) dx = g(x)e ∫ f(x) dx
dx
1
3 dy e ∫ f(x) dx + f(x) e ∫ f(x) dx 2 4 ∫ f(x) dx
y = g(x) e
dx
Note that the left hand side is the differentiation of ye ∫ f(x) dx .

2
d d d
i.e. y e ∫ f(x) dx = 1 y e ∫ f(x) dx + y 1 e ∫ f(x) dx 2
dx dx dx
dy
= 1 2 e ∫ f(x) dx + y 1f(x) e ∫ f(x) dx 2
dx
d
4
\ 3 dx 1ye ∫ f(x) dx 2 = g(x)e ∫ f(x) dx
Integrating both sides with respect to x:
∫1 dx ye ∫ f(x) dx 2 dx = g(x)e ∫ f(x) dx dx
d
∫

∫
\ ye ∫ f(x) dx = g(x)e ∫ f(x) dx dx + C
4 ∫ g(x)e ∫ f(x) dx dx
\ y = + Ce –∫ f(x) dx
e ∫ f(x) dx

Example 8

dy
2
Find the general solution to the differential equation x + 2x = y.
dx
dy
2
Solution: x + 2x = y
dx dy
Arrange the differential equation to the form dx + f(x)y = g(x).
dy y
Divide by x: + 2x =
dx x
dy – 1 x 2 y = –2x
1
dx
dy + – 1 y = –2x
1 2
dx x
1
\ f(x) = – and g(x) = –2x
x
1
–—dx
\ The integrating factor is e ∫ x = e –ln x = e ln x –1 = x . (Notes: e ln a = a)
–1

134

04 STPM Math(T) T2.indd 134 28/01/2022 5:44 PM