This document provides an overview of first order ordinary differential equations. It defines order and degree, and describes six main methods for solving first order ordinary differential equations: variable separable, homogeneous, exact, non-exact, linear, and non-linear. It also discusses applications of first order ordinary differential equations to computer science/IT engineering, numerical analysis, and real-world problems like cooling/warming, population growth, and falling objects.

1. Presentation on
First Order Ordinary Differntial
Equation
 Presented By :
• Deep Dalsania
(160350116002)
• Jhanvi Ghediya
(160350116003)
• Rakesh Talaviya
(160350116010)
• Drashti Bangoriya
(160350116001)
• Bhakti Tank
(160350116011)
• Subject Name: Advanced
Engineering Mathematics
• Subject Code: 2130002
• Submitted To:
Prof. Dipesh Bhogayata

2. Order:-
The order of a differential equation is defined as the order the highest
derivative involve in the differential equation.
Degree:-
The degree of a differential equation is defined as the degree of the highest
derivative involved in differential equation.
Example:-
For this equation order and degree both are 1.
xydxdy  2/

3. Method of solution of ordinary differential
equation of first order and first degree
• There are mainly six methods
1. Variable separable
2. Homogeneous
3. Exact
4. Non-Exact
5. Linear
6. Non-Linear

4. Variable Separable
• Case 1:-
Taking Integration
where c is any arbitrary constant.
0)()(  dyyNdxxM
   cdyyNdxxM )()(

5. Case 2:-
0)()()()( 2211  dyyfxfdxygxf
0)(/)()(/)( 1221  dyygygdxxfxf
Taking Integration
0)(/)()(/)( 1221   ygygdxxfxf

6. Case 3:-
)(/ cbyaxfdxdy 
then
 






dxtbfadt
dxtbfadt
tbfadxdt
tfadxdtb
adxdtbdxdy
dxdtdxbdya
tcbyax
)(/
)(/
)(/
)()/(/1
)/(/1/
//

7. Homogeneous
A function f(x,y) in two variables a and y is said to be
Homogeneous Function of degree n. if
Homogeneous Function:-
),(),( yxfttytxf n

The differential equation is said to be homogeneous if
M(x,y) and N(x,y) both are homogenous function of same degree.
0),(),(  dyyxNdxyxM
0),(),(  dyyxNdxyxM
0),(),(  dyyxNdxyxM
),(/),(/ yxNyxMdxdy 
Let
dxdvvdxdy
vxy
// 


8. Application of first order ordinary
differential equation related to Computer and
IT Engineering
• Numerical solutions and simulations. One reason computers are so useful is that
they solve problems that do not have an analytical solution or where it is difficult
to find one.
• The world around us is governed by differential equations, so any scientific
computing will generally rely on a differential equation and its numerical solution.
Numerical solutions

9. • The study of using computers to solve differential equations generally belongs to
Numerical analysis, not Computer Science.
• The use of differential equations to understand computer hardware belongs to
applied physics or electrical engineering.
• If not most, optimization algorithms in CS are based on solving discrete versions of
differential equations. Gradient descent to name one.
• Many discrete filters act as approximations to analog filters and you can always
derive differential equations to analyze their performance in the continuous domain.
Computer software

10. • The following tables provide a comparison of computer algebra systems (CAS).
• A CAS may include a user interface and graphics capability
• A large library of algorithms, efficient data structures and a fast kernel.
• A computer algebra system (CAS) is a software program that allows computation
over mathematical expressions in a way similar to the traditional manual
computations of mathematicians and scientists.
• A user interface allowing to enter and display mathematical formulas,
• A programming language and an interpreter (the result of a computation has
commonly an un-predictable form.
computer algebra systems

11. Application of first order ordinary
differential equation to Real world
• There are many applications of first order differential equations to real
world problems.
1) Cooling/ and warming law
2) Population growth and decay
3) Harvesting renewable natural resources
4) Prey and predator
5) A Falling object with air resistance