FIRST ORDER DIFFERENTIAL EQUATIONS 2

th
14 February 2026




First Order Differential Equations 2
Peter J Omale


Contents
 Linear differential equations.
 Equations reducible to the linear equation (Bernoulli’s equation).
 Exact differential equations.

Performance objectives
At the end of this programme, students should be able to:
 Solve linear differential equations.
 Reduce Bernoulli’s equation to linear form
 Solve Bernoulli’s equation.
 Show that a given differential equation is exact.
 Solve exact differential equations.




Linear differential equations
A linear differential equation is any equation of the form
dy
+ Py = Q
dx
Where P and Q are functions of x or constants.


Example 1
dy 3x
Solve - 5y = e
dx
dy
Compare with + Py = Q
dx
3x
P = -5 Q=e
∫Pdx=∫-5dx = -∫5dx=-5x


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1

, th
14 February 2026


∫Pdx
Integrating factor IF necessary for solving this problem is given by e .
. ∫Pdx -5x
. . IF = e =e



The solution is

∫
y.IF = Q .IF. dx
-2x
e
= ∫e ∫
-5x 3x -5x -2x
y.e .e dx = e dx = +c
-2
-2x -2x
e e 1
=- + c = c- = c - 2x
2 2 2e
y 1
5x = c - 2x
e 2e
5x
Multiply through by e
3x
e 5x
y = ce -
2


Example
dy 4
Solve x - 2y = x
dx
dy 4
x - 2y = x
dx
Divide through by x
dy 2 3
- y=x
dx x
Compare with
dy
+ Py = Q
dx
2 3
P =- Q=x
x
2 1
∫Pdx = ∫- x dx = -2∫ x dx = -2lnx = lnx -2



∫pdx lnx
-2
-2 1
I.F= e =e =x = 2
x


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2