This slide is about LINEAR DIFFERENTIAL EQUATION & BERNOULLI`S EQUATION. It is one of the important parts of mathematics. This slide will help you to understand the basis of these two parts one Linear Differential Equation and other Bernoulli`s equation.

1. LINEAR DIFFERENTIAL EQUATION &
BERNOULLI`S EQUATION

2. LINEAR DIFFERENTIAL EQUATION
› A differential equation is linear, if
 Dependent variable and its derivatives are of degree one
 Coefficients of a term does not depend upon dependent
variable.
• A linear first order equation is an equation that can be
expressed in the form
d𝑦𝑦
d𝑥𝑥
+ 𝑝𝑝 𝑥𝑥 𝑦𝑦 = 𝑄𝑄 𝑥𝑥
Where P and Q are function of x

3. WAY OF SOLUTION LDE
Integrating Factor
= e∫ 𝑝𝑝 𝑥𝑥 d𝑥𝑥
The general form for solution
𝑦𝑦e∫ 𝑝𝑝 𝑥𝑥 d𝑥𝑥
= �𝑄𝑄e∫ 𝑝𝑝 𝑥𝑥 d𝑥𝑥
d𝑥𝑥 + 𝑐𝑐

4. SOLUTION OF A PROBLEM
Solution of
d𝑦𝑦
d𝑥𝑥
+ 𝑦𝑦 sec 𝑥𝑥 = tan 𝑥𝑥
I.F = e∫sec 𝑥𝑥
= eln sec 𝑥𝑥+tan 𝑥𝑥
= sec 𝑥𝑥 + tan 𝑥𝑥

5. SOLUTION OF A PROBLEM
𝑦𝑦e∫ 𝑝𝑝 𝑥𝑥 d𝑥𝑥
= �𝑄𝑄e∫ 𝑝𝑝 𝑥𝑥 d𝑥𝑥
d𝑥𝑥 + 𝑐𝑐
or, 𝑦𝑦 sec 𝑥𝑥 + tan 𝑥𝑥 = ∫ tan 𝑥𝑥 sec 𝑥𝑥 + tan 𝑥𝑥 d𝑥𝑥 + 𝑐𝑐
= ∫ tan 𝑥𝑥 sec 𝑥𝑥 + tan 𝑥𝑥 tan 𝑥𝑥 d𝑥𝑥 + 𝑐𝑐
= sec 𝑥𝑥 + tan 𝑥𝑥 −𝑥𝑥 + 𝑐𝑐
Since 𝑦𝑦 = 1 −
𝑥𝑥−𝑐𝑐
sec 𝑥𝑥+tan 𝑥𝑥

6. BERNOULLI`S EQUATION
The general form of Bernoulli's equation is
d𝑦𝑦
d𝑥𝑥
+ 𝑝𝑝 𝑥𝑥 𝑦𝑦 = 𝑄𝑄 𝑥𝑥 𝑦𝑦 𝑛𝑛
where n not equal to 0,1 and P and Q are function of x or
constant.
Dividing this equation by 𝑦𝑦 𝑛𝑛
it can be easily reduced to linear
differential equation.

7. BERNOULL`S EQUATION
Integrating Factor
= 𝑒𝑒∫ 1 − 𝑛𝑛 𝑝𝑝 𝑥𝑥 𝑑𝑑𝑥𝑥
Solution form of Bernoulli's equation
𝑦𝑦1−𝑛𝑛
𝐞𝐞∫ 1−𝑛𝑛 𝐩𝐩(𝐱𝐱) 𝐝𝐝𝑥𝑥
= � 1 − 𝑛𝑛 𝑄𝑄𝐞𝐞∫ 1−𝑛𝑛 𝑝𝑝 𝑥𝑥 𝐝𝐝𝑥𝑥
𝐝𝐝𝑥𝑥 + 𝑐𝑐

8. SOLUTION OF BERNOULLI`S EQUATION
Given equation
𝑑𝑑𝑑𝑑
𝑑𝑑𝑑𝑑
+
1
𝑥𝑥
y= 𝑥𝑥𝑦𝑦2
… … … … . . 1
Which is a Bernoulli's differential equation.
Here,𝑝𝑝 𝑥𝑥 =
1
𝑥𝑥
, 𝑄𝑄 𝑥𝑥 = 𝑋𝑋
And n=2
Integrating factor =𝑒𝑒∫ 1−𝑛𝑛 𝑝𝑝 𝑥𝑥 𝑑𝑑𝑑𝑑
=𝑒𝑒∫ 𝑝𝑝(𝑥𝑥)
1
𝑥𝑥
𝑑𝑑𝑑𝑑
=𝑒𝑒− ∫
1
𝑥𝑥
𝑑𝑑𝑑𝑑
=𝑒𝑒−𝑙𝑙 𝑙𝑙𝑙𝑙
=𝑒𝑒𝑙𝑙 𝑙𝑙 𝑥𝑥−1
=
1
𝑥𝑥

9. Solution of (1)will be..
𝑦𝑦1−𝑛𝑛
𝑒𝑒∫ 1−𝑛𝑛 𝑝𝑝 𝑥𝑥 𝑑𝑑𝑑𝑑
= � 1 − 𝑛𝑛 𝑄𝑄 𝑒𝑒∫ 1−𝑛𝑛 𝑝𝑝 𝑥𝑥 𝑑𝑑𝑑𝑑
𝑑𝑑𝑑𝑑 + 𝑐𝑐
or, 𝑦𝑦1−2 1
𝑥𝑥
= ∫ 1 − 2 𝑥𝑥
1
𝑥𝑥
dx + c
or
1
𝑦𝑦
1
𝑥𝑥
= -∫ 𝑑𝑑𝑑𝑑 + 𝑐𝑐
or
1
𝑥𝑥𝑥𝑥
= -∫ 𝑑𝑑𝑑𝑑 + 𝑐𝑐
or
1
𝑥𝑥𝑥𝑥
= −x + c
(Ans)

10. Name: Md.Touhidul Islam Shawan
Studying B.Sc. In Computer
Science and Engineering at
Daffodil International University
FB:
www.facebook.com/touhidulshao
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This slide is created by me.
Anyone can use this if you need.