Differential equation and its order and degree

The order of a differential equation is determined by the highest-order derivative; the degree is determined by the highest power on a variable. The higher the order of the differential equation, the more arbitrary constants need to be added to the general solution

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, + +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, + Partial differential equations: A partial differential equation is a differential equation that involves partial derivatives. For example,  

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 , 3 y′′+2 ln(x)y′+ y= 3x cosx Non-linear differential equations: A nonlinear partial differential equation is a partial differential equations with  non linear terms. For example , 4yy''' - x 3 y' + cos y = e 2x  

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: . 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: x+ + .  

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 = 1 1 + 6y = 0 2 1 + = 0 3 1 y =0 1 2 Differential Equation Order Degree 1 1 2 1 3 1 1 2

10. Example Differential Equation Order Degree + 1 Not Define Differential Equation Order Degree 1 Not Define Reason : Not a polynomial in y   = 2 2 2 2 Reason : We need to remove the radical sign + =  

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. = 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 1 st 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 T o 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.

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