This document discusses differential equations and their solutions. It defines ordinary and partial differential equations. Linear differential equations have dependent variables and derivatives of degree one, and coefficients that do not depend on the dependent variable. The method of integrating factors can be used to solve linear differential equations. Bernoulli's equations can be reduced to linear equations by dividing through by the power of the dependent variable. An example problem demonstrates solving a Bernoulli's equation using the integrating factor method.

1. Title Layout
Subtitle

2. ABDULLAH AL AMIN

3. DIFFERENTIL EQUATION
An equation which involves unknown function of one or several
variables that relates the values of the function itself and its
derivatives of various orders.
 Ordinary differential equation (ode) : not involve partial
derivatives
 Partial differential equation (pde) : involves partial derivatives

4. 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
ⅆ𝑦
ⅆ𝑥
+ 𝑝 𝑥 𝑦 = 𝑄 𝑥
Where P and Q are function of x

5. MD. TOUHIUL ISLAM SHAWAN

6. WAY OF SOLUTION LDE
Integrating Factor
= ⅇ 𝑝 𝑥 ⅆ𝑥
The general form for solution
𝑦ⅇ 𝑝 𝑥 ⅆ𝑥
= 𝑄ⅇ 𝑝 𝑥 ⅆ𝑥
ⅆ𝑥 + 𝑐

7. SOLUTION OF A PROBLEM
Solution of
ⅆ𝑦
ⅆ𝑥
+ 𝑦 sⅇc 𝑥 = tan 𝑥
I.F = ⅇ sec 𝑥
= ⅇln sec 𝑥+tan 𝑥
= sⅇc 𝑥 + tan 𝑥

8. SOLUTION OF A PROBLEM
𝑦ⅇ 𝑝 𝑥 ⅆ𝑥
= 𝑄ⅇ 𝑝 𝑥 ⅆ𝑥
ⅆ𝑥 + 𝑐
= 𝑦 sⅇc 𝑥 + tan 𝑥 = tan 𝑥 sⅇc 𝑥 + tan 𝑥 ⅆ𝑥 + 𝑐
= tan 𝑥 sⅇc 𝑥 + tan 𝑥 tan 𝑥 ⅆ𝑥 + 𝑐
= sⅇc 𝑥 + tan 𝑥 −𝑥 + 𝑐
Since 𝑦 = 1 −
𝑥−𝑐
sec 𝑥+tan 𝑥

9. MD.NOUSHAD ALI

10. BERNOULL`S EQUATION
The general form of Bernoull`s equation is
ⅆ𝑦
ⅆ𝑥
+ 𝑝 𝑥 𝑦 = 𝑄 𝑥 𝑦 𝑛
where n not equal to 0,1 and P and Q are function of x or
constant.
Dividing this equation by 𝑦 𝑛
it can be easily reduced to
linear differential equation.

11. BERNOULL`S EQUATION
Integrating Factor
= ⅇ 1 − 𝑛 𝑝 𝑥 ⅆ𝑥
Solution form of Bernoull`s equation
𝑦1−𝑛ⅇ 1−𝑛 𝐩(𝐱) ⅆ𝑥
= 1 − 𝑛 𝑄ⅇ 1−𝑛 𝑝 𝑥 ⅆ𝑥
ⅆ𝑥 + 𝑐

12. MD SHAKAWAT HOSSAIN

14. AKASH CHANDRA DAS

15. Example
Given equation
ⅆ𝑦
ⅆ𝑥
+
1
𝑥
y= 𝑥𝑦2
… … … … . . 1
Which is a Bernoulli's differential equation.
Here,𝑝 𝑥 =
1
𝑥
, 𝑄 𝑥 = 𝑋
And n=2
Integrating factor =ⅇ 1−𝑛 𝑝 𝑥 ⅆ𝑥
=ⅇ 𝑝(𝑥)
1
𝑥
ⅆ𝑥
=ⅇ−
1
𝑥
ⅆ𝑥
=ⅇ−𝑙𝑛𝑥
=ⅇ 𝑙𝑛𝑥−1
=
1
𝑥

16. Example
Solution of (1)will be..
𝑦1−𝑛
ⅇ 1−𝑛 𝑝 𝑥 ⅆ𝑥
= 1 − 𝑛 𝑄 ⅇ 1−𝑛 𝑝 𝑥 ⅆ𝑥
𝑑𝑥 + 𝑐
or, 𝑦1−2 1
𝑥
= 1 − 2 𝑥
1
𝑥
ⅆx + c
or
1
𝑦
1
𝑥
= - 𝑑𝑥 + 𝑐
or
1
𝑥𝑦
= - 𝑑𝑥 + 𝑐
or
1
𝑥𝑦
= −x + c
(Ans)