The document outlines the classification of differential equations based on their types, order, and degree, explaining ordinary and partial differential equations, linear and non-linear equations, as well as homogeneous and heterogeneous equations. It also details the process of determining the order and degree of a differential equation, provides examples, and discusses various applications in fields such as physics, engineering, and computer science. Additionally, it highlights the significance of differential equations in artificial intelligence and game development.

1. Classification of Differential Equation
And It’s Order And Degree
Presented By:
Monjurul Islam- 181002050
Riyan Sajib Miha-181002053
Jannatul Perdous-181002056
Mahfuzur Rahman Faruk-181002057

2. Content
› Differential equation
› Classification of differential equation
› Order of differential equation
› Degree of differential equation
› Example
› First order differential equation
› Application of differential equation

3. Differential Equation
An equation containing the derivatives of one or
more dependent variables with respect to one
or more independent variable is said to be
differential equation.
For example,
𝜕2 𝑦
𝜕𝑥2 +
𝜕2 𝑦
𝜕𝑡2 +y=0 ,
Y”+Y’+Y=0

4. Classification of Differential
Equation
Differential equations can be divided into several types.
Commonly used distinctions include whether the equation is:
› Ordinary differential equations.
› Partial differential equations.
› Linear differential equations.
› Non-linear differential equations.
› Homogeneous differential equations.
› Heterogeneous differential equations.

5. Classification of Differential Equation
› Ordinary differential equations:
An Ordinary differential equations is a differential
equation that does not involve partial derivatives. For
example,
𝜕2 𝑦
𝜕𝑥2+
𝜕𝑦
𝜕𝑥
= 3𝑥 sin 𝑦
› Partial differential equations:
A partial differential equation is a differential equation that
involves partial derivatives. For example,
𝜕𝑦
𝜕𝑡
+ 𝑥
𝜕𝑦
𝜕𝑥
=
𝑥 + 𝑡
𝑥 − 𝑡

6. Classification of Differential Equation
› Linear differential equations:
A linear differential equation is a differential equations that
is defined by a linear polynomial in the unknown function
and its derivatives. For example ,
3𝑥2y′′+2 ln(x)y′+𝑒 𝑥y= 3x cosx
› Non-linear differential equations:
A nonlinear partial differential equation is a partial
differential equations with non linear terms. For example ,
4yy''' - x3y' + cos y = e2x

7. Classification of Differential Equation
› Homogeneous differential equations:
Involve only derivatives of y and terms involving y, and
they’re set to 0, as in this equation:
𝑑4 𝑦
𝑑𝑥4 𝑥 +
𝑑2 𝑦
𝑑𝑥2 + 𝑦2
= 0.
› Heterogeneous differential equations:
are the same as homogeneous differential equations,
except they can have terms involving only x (and
constants) on the right side, as in this equation:
𝑑2 𝑦
𝑑𝑥2 x+
𝑑𝑦
𝑑𝑥
+𝑦2
= 6𝑥 + 3.

8. Differential Equations Order & Degree
Order –
1. Pick the dependent variable of the differential equation.
2. It’s highest order is the order of the differential equation.
Degree –
1. The differential equation must be a polynomial equation
derivatives.
2. Highest power of the highest order involved is the degree of
the differential equation.

9. Example
Differential Equation Order Degree
𝑑𝑦
𝑑𝑥
= 𝑒 𝑥 1 1
𝑑2 𝑦
𝑑𝑥2 + 6y = 0 2 1
𝑑3 𝑦
𝑑𝑥3 + 𝑥2 𝑑2 𝑦
𝑑𝑥2
3
= 0
3 1
𝑑𝑦
𝑑𝑥
2
+
𝑑𝑦
𝑑𝑥
− 𝑠𝑖𝑛2
y =0 1 2

10. Example
Differential Equation Order Degree
𝑑𝑦
𝑑𝑥
+ sin
𝑑𝑦
𝑑𝑥
= 0 1 Not Define
Reason : Not a polynomial in y′
𝑑2 𝑦
𝒅𝑥2
2
+
𝑑𝑦
𝑑𝑥
3
= y + sin 𝒙
2 2
Reason : We need to remove the radical sign
𝑑2 𝑦
𝑑𝑥2
2
+
𝑑𝑦
𝑑𝑥
3
= y + sin 𝒙 2

11. First order differential equation
A differential equation which have one or more dependent
variables with respect to a single independent variable and
which has only one order derivatives, is called a 1 st order
ordinary differential equation.
𝑑𝑦
𝑑𝑥
=
𝑥2+𝑦2
𝑥+𝑦
Here y is a dependent variable and x is a independent variable
Types of first order ODE :
 Separable equation
 Homogenous equation
 Non Homogeneous equation
 Exact equation
 Linear equation

12. Separable equations

13. Exact equation
Exact Equation is called an exact differential equation if
there exists a function of two variables u(x, y) with
continuous partial derivatives such that
du(x, y) = P(x, y)dx + Q(x, y)dy.
The general solution of an exact equation is given by
u(x, y)=C,
where C is an arbitrary constant.

14. Applications of 1st order ordinary differential
equation :
› Cooling/Warming Law (use in physics)
› Population Growth and Decay (in stat..)
› Radio-Active Decay and Carbon Dating
› Mixture of Two Salt Solutions(in chemistry)
› Series Circuits (in physics)
› Survivability with AIDS (in medicine)
› Draining a tank (in engineering)
› Economics and Finance ( in economics)
› Drug Distribution in Human Body ( in biology)

15. APPLICATION OF DIFFERENTIAL EQUATION
› Creating Software’s
› Computer Programming
› Creating Games
› Artificial Intelligence
› Modeling Electrical Circuit

16. CREATING SOFTWARE
A computer can be an extremely valuable tool in the study of differential equation.
There exists extremely powerful and general software package that can perform a
wide variety of mathematical operations. The use of differential equation to
understand computer hardware belongs to applied physics or electrical engineering.

17. IN COMPUTER PROGRAMMING
To create a program many types of differential equation is needed. Specially in
logic programming ordinary differential equation is used to make a program.

18. CREATING GAMES
Differential Equation is used to model the velocity of the gaming character.
Differential equation is an essential tool for describing the nature of the
physical universe and naturally also an essential part of models for computer
graphics and vision.

19. ARTIFICIAL INTELLIGENCE
Both ordinary and partial differential equation is used to create artificial
intelligence software. To make AI scene detector software and AI robots
differential equation is essential.

20. MODELING ELECTRICAL
CIRCUIT
Another application of first order differential equations arises in the modeling
of electrical circuits.

21. Any Question

22. Thank You