In this presentation you can learn about the linear differential equation and bernoullis equations.

1. GANDHINAGAR INSTITUTE OF
TECHNOLOGY
MECHANICAL DEPARTMENT
ADVANCE ENGINEERING MATHEMATICS
LINEAR DIFFERENTIAL EQUATION AND BERNOULLIS
EQUATIONS

2. CONTENTS :-
• Introduction
• Integrating Factor
• Linear Differential Equation
• Bernoulli’s Equation

3. INTEGRATING FACTOR :-
• Sometimes, a differential equation is not exact but can be made by
exact by multiplying with a suitable function. This function is known
as Integrating Factor(IF).

4. LINEAR DIFFERENTIAL EQUATION :-
• In each term in a differential equation including the derivative in
linear in the term of the dependent variable then the equation is
linear.
• A Differential Equation in the form of
ⅆ𝑦
ⅆ𝑥
+ 𝑝𝑦 = 𝑄 …(i)
Where P and Q are the function of 𝑥, is called a linear differential equation
and linear is 𝑦.

5. • Solve the equation (i) to obtain the Integrating Factor (I.F.)….
IF = ⅇ 𝑃 ⅆ𝑥
• Multiplying equation (i) by IF,
ⅇ 𝑃 ⅆ𝑥 ⅆ𝑦
ⅆ𝑥
+ Pⅇ 𝑃 ⅆ𝑥
𝑦 = Qⅇ 𝑃 ⅆ𝑥
ⅆ
ⅆ𝑥
[ⅇ 𝑃 ⅆ𝑥
𝑦] = Qⅇ 𝑃 ⅆ𝑥
Integration W.R.T 𝑥
ⅇ 𝑃 ⅆ𝑥
𝑦 = Qⅇ 𝑃 ⅆ𝑥
dx + c
(IF) 𝑦 = (𝐼𝐹)Q + c …(ii)
Equation (ii) is known as a differential equation.

6. EXAMPLE :- ⅆ𝑦
ⅆ𝑥
+ 3 𝑦
𝑥 = 𝑆ⅈ𝑛 𝑥
𝑥3
• Solution:- The equation is linear in 𝑦.
P =
3
𝑥
, Q = 𝑆ⅈ𝑛 𝑥
𝑥3
IF = 𝑒
3
𝑥 𝑑𝑥
= 𝑒3log𝑥
= ⅇlog 𝑥3
= 𝑥3
Here,
𝑥3 𝑦 = 𝑥3 sin 𝜒
𝑥3 ⅆ𝑥 + 𝐶
= sin 𝑥 ⅆ𝑥 + 𝐶

7. - cos 𝑥 + 𝐶
Y = -
Cos 𝜒
𝑥3 + 𝐶

8. BERNOULLI’S EQUATION :-
• The Equation of the form
ⅆ𝑦
ⅆ𝑥
+ 𝑝𝑦 = 𝑄𝑦 𝑛 … (i)
• P and Q are the function of x or constants is nonlinear
equation, known as Bernoulli’s Equation.
• Divide the equation (i) by 𝑦 𝑛
,
1
𝑦 𝑛
ⅆ𝑦
ⅆ𝑥
+ 𝑝
𝑦 𝑛−1 = 𝑄 …(ii)

9. Let ,
1
𝑦 𝑛−1 = 𝑣
1−𝑛
𝑦 𝑛
ⅆ𝑦
ⅆ𝑥
=
𝑑𝑣
𝑑𝑥
1/𝑦 𝑛 ⅆ𝑦
ⅆ𝑥
= 1/(1 − 𝑛)
𝑑𝑣
𝑑𝑥
Substitute in equation (ii)
1
1−𝑛
𝑑𝑣
𝑑𝑥
+ 𝑝𝑣 = 𝑄
𝑑𝑣
𝑑𝑥
+ 1 − 𝑛 𝑝𝑣 = 𝑄

10. E𝐗𝐀𝐌𝐏𝐋𝐄 ∶
𝑑𝑦
𝑑𝑥
+
2𝑦
𝑥
= 𝑦2
𝑥2
Now
1
𝑦2
𝑑𝑦
𝑑𝑥
+
1
𝑦
.
2
𝑥
= 𝑥2
Let
1
𝑦
= 𝑣 −
1
𝑦2
𝑑𝑦
𝑑𝑥
=
𝑑𝑣
𝑑𝑥
… (i)
Substitute in equation (i),
−
𝑑𝑣
𝑑𝑥
+
2
𝑥
𝑣 = 𝑥2
𝑑𝑣
𝑑𝑥
−
2
𝑥
𝑣 = −𝑥2
…(ii)
Here, 𝑝 =
2
𝑥
, 𝑄 = −𝑥2

11. 𝐼𝐹 = 𝑒 −2
𝑥 𝑑𝑥
= 𝑒−2log𝑥
= ⅇlog 𝑥−2
=
1
𝑥2
1
𝑥2=
1
𝑥2 − 𝑥2 𝑑𝑥 + 𝑐
= − 𝑑𝑥 + 𝑐
= −𝑥 + 𝑐
𝑣 = −𝑥3 +𝑐𝑥2
1
𝑦
= − 𝑥3
+𝑐𝑥2

12. THANK YOU …