Page 29 - Pra U STPM 2022 Penggal 2 - Mathematics

Mathematics Semester 2 STPM Chapter 4 Differential Equations

dy y
So + x y = – reduces to
2
dx x
1 dv v 2 v 1 v
x 1 x 2
1 x dx – x 22 + x 1 x 2 = –
1 dv – v + vx = – v

x dx x 2 x 2
1 dv + vx = 0

x dx

\ dv = –vx which is the form where the variables are separable.
2
dx
Separating the variables and integrating with respect to x.
∫
2
\ dv = – x dx
∫ v
\ ln v = – x 3 2Sdn Bhd. All Rights Reserved.
+ A
3
Substituting v = yx
\ ln yx = – x 3 + A is the general solution.
3

Example 11

2
3
dy y – x y
Solve the differential equation = .
dx x + xy 2
3
Penerbitan Pelangi
4
3
dy y – x y
Solution: =
3
dx x + xy 2
Substituting y = vx,
3
v + x dv = v – v dy = v + x dv
dx 1 + v 2 dx dx
3
x dv = v – v – v
dx 1 + v 2
x dv = – 2v
dx 1 + v 2

∫
∫ 1 + v 2 dv = – dx
2
v
x
ln | v | + 1 v = –2 ln | x | + C C is an arbitrary constant.
2
2
2
ln | vx | + 1 v = C
2
2
ln | yx | = – y 2 + C
2x 2
y 2
–—
yx = Ae 2x where A = e C

2
137

04 STPM Math(T) T2.indd 137 28/01/2022 5:44 PM