This document provides an introduction to ordinary differential equations (ODEs). It defines ODEs as differential equations containing functions of one independent variable and its derivatives. The document discusses some key concepts related to ODEs including order, degree, and different types of ODEs such as variable separable, homogeneous, exact, linear, and Bernoulli's equations. Examples of each type of ODE are provided along with the general methods for solving each type.

1. Introduction to Ordinary Differential
Equations
Presented by
Dnyaneshwar Pardeshi [182070006]
M.Tech (Control System)
Under The Guidance
Of
Dr. Surendra Bhosale

2. Introduction
 Differential equations are introduce in different fields and
its importance appears not only in mathematics but also in
Engineering , Natural science ,Chemical science , Medicine,
Ecology and Economy.
 In mathematics, an ordinary differential equation is a
differential equation containing one or more functions of
one independent variable and its derivatives. The term
ordinary is used in contrast with the term partial differential
equation which may be with respect to more than one
independent variable.

3. Notation and Definitions
• Differential equation:
Equations, which are composed of an unknown
function and its derivatives.
1. Ordinary Differential equation: When the
function involves one independent variable, the
equation is called an ordinary differential equation
(or ODE).
2. Partial Differential equation: Differential
equation involving two or more independent
variables and its differentials.

4. Order :
The order of highest derivative used in a differential
equation
Degree :
Power of a highest derivative which is free from rationals
& radical in the differential equation.
02
2

dt
dy
dt
yd.
2nd order, 1st degree
3
3
dt
xd
x
dt
dx

3rd order, 1st degree

5. Solutions of First Order Differential Equations
1. Variable Separable Equations
2. Homogeneous & Non homogeneous Equation
3. Exact & Non-Exact Equations
4. Linear Equations
5. Bernoulli’s Equations

6. 1- Variable Separable Equations
General form of differential equation is
By seperating variable we get,
By integrating we find the solution of this equation.
0),(),(  dyyxhdxyxf
0)()()()( 2121  dyyhxhdxyfxf
0
)(
)(
)(
)(
2
1 2
1
 dy
yf
yh
dx
xh
xf

7. 2- Homogeneous Equation
The condition of homogeneous function is
where ‘n’ is degree of homogeneous function.
Then
Is the homogeneous differential equation if ‘f’ & ‘g’ are
homogeneous function of same degree.
Can be solved by reducing it into separable form by
substitution i.e.
),(),( yxfyxf nn
 
),(
),(
yxg
yxf
dx
dy

vxy 
dx
dv
xv
dx
dy


8. 3- Exact Equations
The required condition for equation to be exact
is
and its general solution is
[ is constant] [Terms free from ]
0),(),(  dyyxNdxyxM
x
N
y
M





cdyyxNdxyxM   ),(),(
y x

9. 4. Linear Equations
The standard form of linear DE is
...linear in y
The integral factor that convert Linear Equations
to exact equation is
The general solution is
)()( xQyxP
dx
dy


Pdx
eFI.
cdxFIQFIy   ).().(

10. 5. Bernoulli’s Equations
I. Divide Bernoulli Equation over
II. Put then
This is linear equation & its solution as we told before
n
yxQyxP
dx
dy
 )()(
n
y
n
yz 
 1
)()(
1 1
xQyxP
dx
dy
y
n
n
 
dx
dy
ydx
dz
n n
1
)1(
1


    )(1)(1 xQnzxPn
dx
dz


11. THANK YOU