Page 28 - Pra U STPM 2022 Penggal 2 - Mathematics

Mathematics Semester 2 STPM Chapter 4 Differential Equations

Exercise 4.3

1. Solve the differential equations
2 dy
(a) (1 + x ) + xy = x + x 3 (b) dy + 3y = x
dx dx
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dy dy
2
(c) x + y = 2x (d) = 2x – y
dx dx
dy
(e) 3 + 15y = 24e 3x
dx
2. Solve the differential equations given the boundary conditions:
dy 2 dy
(a) x – 3y = x(1 + x ); x = 1, y = 1 (b) (1 + x ) + xy = x ; x = 1, y = 0
3
dx dx
dy dy
–x
(c) + 3y = 2e ; x = 0, y = 1 (d) – 2xy = x ; x = 1, y = 1
dx dx

4.4 Transformations Differential Equations

Some first order differential equations which are not of the form where the variables are separable can be reduced
to the separable forms by suitable substitutions.

4
Example 10

dy y
2
Solve the differential equation + x y = – .
dx x
dy y
2
Solution: The variables of the differential equation dx + x y = – are not separable.
x
Let y = v x

Differentiating with respect to x:
dy = x v + v d x –1
–1 d
dx dx dx
dy = x + v(–x )
–1 dv
–2
dx dx
dy 1 dv v
\ = –
dx x dx x 2

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