Page 31 - Pra U STPM 2022 Penggal 2 - Mathematics

Mathematics Semester 2 STPM Chapter 4 Differential Equations

1 dx
∫
Multiplying both sides with the integrating factor e x :
du 1 u 1 1
∫
∫
∫
e x dx + e x dx = –1(e x dx )
dx x
du
x + u = –x
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dx
d
(ux) = –x
dx
∫
\ ux = –x dx + A
\ ux = – x 2 + A
2
Substituting u = 1 ,
y 2
\ x = – x 2 + A
y 2 2

2
\ y = 2x
–x + 2A
2
2x
2

\ y = ± 
2
–x + 2A
Exercise 4.4

1. Solve the differential equations 4
dy x + y dy x – 2y
(a) = (b) =
dx x dx x
2
dy x + 2y 2 dy
2
(c) = (d) xy = x + y 2
dx xy dx
dy dy y – yx 2
3
(e) xy = x – y 2 (f) =
2
dx dx x + y x
3
2
dy xy
2. Show that, by substituting y = vx, the differential equation dx = x + y 2 can be transformed into
2
x dv + v 3 = 0. Hence, find the particular solution if y = 2 when x = 1.
dx 1 + v 2
dy y – x + 1
3. Using the substitution y – x = z, show that the differential equation dx = y – x + 5 can be transformed
into dz = – 4 . Hence, solve the differential equation.
dx z + 5
dy x + y + 2
4. Using the substitution z = x + y, solve the differential equation = .
dx x + y + 1
dy 2x – y – 1
5. Solve the differential equation = by using the substitution v = 2x – y.
dx 2x – y + 3

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04 STPM Math(T) T2.indd 139 28/01/2022 5:44 PM