Page 22 - Pra U STPM 2022 Penggal 2 - Mathematics

Mathematics Semester 2 STPM Chapter 4 Differential Equations

Exercise 4.1

1. State the order and degree of the following differential equations.
2
2
(a) x 2 d y + x dy + 4y = 0 (b) 2x d y + (sin x) dy – y = cos x
dx 2 dx dx 2 dx
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2
dy
(c) 1 2 2 + x = y (d) d y 2 – x = dy
dx
dx
dx
2
3
dy
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(e) 1 d y 2 + x d y 2 1 2 5 + y = cos x
4
+
dx
dx
dx
2. State the degree of the following differential equations.
5
—
2
dy d y 2 3
3
(a) = 1 + 1 22 4
dx dx
dy
(b) 1 y – x dy 2 2 = 1 + 1 2 2
dx
dx
d s  2
2
ds
(c) 2  1 2
= 5 +
dt dt
3. Show that y = 3x + 2A is the general solution of the differential equation x dy + y = 6x. Hence, obtain
x
dx
dy
the particular solution of the differential equation x + y = 6x if y = 1 when x = 1.
dx
dy
4. Show that y + 2 = Ax + x is the general solution of the differential equation x – y = 2 + x . Hence,
2
2
dy 2 dx
4 obtain the particular solution of the differential equation x dx – y = 2 + x , given that y = 1 when
x = 1.
4.2 First Order Differential Equations with
Separable Variables
Generally, a first order differential equation can be written as
dy
= f(x, y) .....................................................
dx
In certain cases, function f(x, y) can be written as
.
f(x, y) = u(x) v(y) ..............................................
where u(x) and v(y) are functions of x and y respectively.
By substituting equation  into equation ,
dy
= u(x) v(y)
.
dx
Equation  is a differential equation with separable variables if it can be written as
dy = u(x) dx
v(y)

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