LINEAR DIFFERENTIAL EQUATION & BERNOULLI`S EQUATION

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
›Adifferentialequationis linear,if
Depend entvariable an d itsderivativesare ofdegree one
Coefficientsof atermdoes not dependupondepend ent
variable.
•Alinear firstorderequation is an equation thatcanbe
expressedintheform
d????????????
d????????????
+????????????????????????????????????=????????????????????????
Where P and Q are function of x

3. WAY OF SOLUTION LDE
Integrating Factor
= e
∫????????????????????????d????????????
The general form for solution
????????????e
∫????????????????????????d????????????
=�????????????e
∫????????????????????????d????????????
d????????????+????????????

4. SOLUTION OF A PROBLEM
Solution of
d????????????
d????????????
+????????????sec????????????=tan????????????
I.F = e
∫sec????????????
= e
lnsec????????????+tan????????????
= sec????????????+tan????????????

5. SOLUTION OF A PROBLEM
????????????e
∫????????????????????????d????????????
=�????????????e
∫????????????????????????d????????????
d????????????+????????????
or, ????????????sec????????????+tan????????????=∫tan????????????sec????????????+tan????????????d????????????+????????????
= ∫tan????????????sec????????????+tan????????????tan????????????d????????????+????????????
= sec????????????+tan????????????−????????????+????????????
Since ????????????=1−
????????????−????????????
sec????????????+tan????????????

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

7. BERNOULL`S EQUATION
Integrating Factor
= ????????????
∫1−????????????????????????????????????????????????????????????
Solution form of Bernoulli's equation
????????????
1−????????????
????????????
∫1−????????????????????????(????????????)????????????????????????
=�1−????????????????????????????????????
∫1−????????????
????????????
????????????????????????????????????
????????????????????????+????????????

8. SOLUTION OF BERNOULLI`S EQUATION
Given equation
????????????????????????
????????????????????????
+
1
????????????
y=????????????????????????
2
…………..
1
Which is a Bernoulli'sdifferential equation.
Here,????????????????????????=
1
????????????
,????????????
????????????=????????????
And n=2
Integrating factor =????????????
∫1−????????????????????????????????????????????????????????????
=????????????
∫????????????(????????????)
1
????????????
????????????????????????
=????????????
−∫
1
????????????
????????????????????????
=????????????
−????????????????????????????????????
=????????????
????????????????????????????????????
−1
=
1
????????????

9. Solution of (1)will be..
????????????
1−????????????
????????????
∫1−????????????????????????????????????????????????????????????
=�1−????????????????????????????????????
∫1−????????????????????????????????????????????????????????????
????????????????????????+????????????
or, ????????????
1−2
1
????????????
= ∫
1−2????????????
1
????????????
dx+c
or
1
????????????
1
????????????
= -∫????????????????????????+????????????
or
1
????????????????????????
= -∫????????????????????????+????????????
or
1
????????????????????????
=−x+c
(Ans)

10. Name: Md.TouhidulIslam 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.