February 14, 2026
First Order Differential Equations 1
Peter. J. Omale
Contents
Order of differential equations
Formation of differential equations
Solution of differential equations by direct integration
Variable separable form of differential equations
Homogeneous differential equations
Performance objectives
At the end of this programme, students should be able to:
State the order of differential equations.
Form differential equations from given functions.
Solve differential equations by direct integration.
Solve differential equations by separating the variables.
Identify homogeneous differential equations.
Solve homogeneous differential equations.
Order of differential equations
Any equation which contains differential coefficient(derivative)is called
differential equation. Examples are given below. The order of a differential
equation is given by the highest derivative that is in the equation
For example
dy
+ 3y = 0 is an equation of order 1
dx
2
dy 3
5 2 - y cos x=0 is an equation of order 2
dx
2
3 dy 2
x y 2 + y tan x = 4x is an equation of order 2
dx
Redistribution of this material by photocopying or any other means is prohibited. 1
matrojek
, February 14, 2026
3 2
dy dy
sin x 3 - cos x 2 = 2y is an equation of order 3
dx dx
Formation of differential equations
We form differential equation by differentiating a given function and
eliminating the arbitrary constants.
Example1
Form differential equation from the function
2
y = 3x + 5x + c where c is an arbitrary constant.
Solution
2
y = 3x + 5x + c
Differentiating this, we get
dy
= 6x + 5
dx
Note that differentiating c gives 0, and so the arbitrary constant is
eliminated.
Example 2
Form the associated differential equation from the following function:
y = A cos x - Bsin x
We differentiate the function and eliminate the constants A and B
y = Acos x-Bsin x
dy
= -A sin x-Bcos x
dx
We differentiate again
2
dy
2 = -Acos x+Bsin x = -(Acos x-Bsin x)
dx
To eliminate A and B, we substitute y for the terms containing them.
2
dy
2 = -y
dx
2
dy
2 + y = 0
dx
Redistribution of this material by photocopying or any other means is prohibited. 2
matrojek
First Order Differential Equations 1
Peter. J. Omale
Contents
Order of differential equations
Formation of differential equations
Solution of differential equations by direct integration
Variable separable form of differential equations
Homogeneous differential equations
Performance objectives
At the end of this programme, students should be able to:
State the order of differential equations.
Form differential equations from given functions.
Solve differential equations by direct integration.
Solve differential equations by separating the variables.
Identify homogeneous differential equations.
Solve homogeneous differential equations.
Order of differential equations
Any equation which contains differential coefficient(derivative)is called
differential equation. Examples are given below. The order of a differential
equation is given by the highest derivative that is in the equation
For example
dy
+ 3y = 0 is an equation of order 1
dx
2
dy 3
5 2 - y cos x=0 is an equation of order 2
dx
2
3 dy 2
x y 2 + y tan x = 4x is an equation of order 2
dx
Redistribution of this material by photocopying or any other means is prohibited. 1
matrojek
, February 14, 2026
3 2
dy dy
sin x 3 - cos x 2 = 2y is an equation of order 3
dx dx
Formation of differential equations
We form differential equation by differentiating a given function and
eliminating the arbitrary constants.
Example1
Form differential equation from the function
2
y = 3x + 5x + c where c is an arbitrary constant.
Solution
2
y = 3x + 5x + c
Differentiating this, we get
dy
= 6x + 5
dx
Note that differentiating c gives 0, and so the arbitrary constant is
eliminated.
Example 2
Form the associated differential equation from the following function:
y = A cos x - Bsin x
We differentiate the function and eliminate the constants A and B
y = Acos x-Bsin x
dy
= -A sin x-Bcos x
dx
We differentiate again
2
dy
2 = -Acos x+Bsin x = -(Acos x-Bsin x)
dx
To eliminate A and B, we substitute y for the terms containing them.
2
dy
2 = -y
dx
2
dy
2 + y = 0
dx
Redistribution of this material by photocopying or any other means is prohibited. 2
matrojek
