APPLICATION OF FIRSTORDER DIFFERENTIAL EQUATIONS
G. Mahata ^{1 }, D.S Raut ^{1}, C. Parida ^{1}, S. Baral ^{1}, S. Mandangi ^{1}
^{1}^{ }Department of Mathematics C. V. Raman College of Engineering, Bhubaneswar752054, India

ABSTRACT 

This study
introduced real life application of first order differential equation. In
this paper We basically discussed about different types of differential
equation and the solution of first order differential equation and
application of first order differential equation in different field of
science and technology. Further, Newton’s law of cooling and orthogonal
trajectory has been incorporated. Study about convective boundary condition
and it is used for increasing the temperature. 

Received 29 August 2022 Accepted 20 September 2022 Published 08 October 2022 Corresponding Author G. Mahata, ganeswar0385@gmail.com
DOI 10.29121/IJOEST.v6.i5.2022.402 Funding: This research
received no specific grant from any funding agency in the public, commercial,
or notforprofit sectors. Copyright: © 2022 The
Author(s). This work is licensed under a Creative Commons
Attribution 4.0 International License. With the
license CCBY, authors retain the copyright, allowing anyone to download,
reuse, reprint, modify, distribute, and/or copy their contribution. The work
must be properly attributed to its author. 

Keywords: Differential
Equations, Orthogonal Trajectory, Radiation 
1. INTRODUCTION
The equation contains one or more than one derivative is known as differential equations or in other words we can say the equation that contain derivative of one or more dependent variable with respect to one or more independent variables. Differential equations have remarkable ability to predict the world around us. They are used in a wide variety of disciplines, from biology, economics, chemistry, and engineering. The differential equation which contains only first order derivative is known as first order differential equation. Basically, we divide differential equation into two categories that are ordinary differential equation and partial differential equation. The ordinary differential equation contains unknown functions of one independent variable whereas the partial differential equation contains unknown function of more than one independent variable. We discussed more about this further in this paper. In this paper we are going to discussed about first order differential equations and its solution and its application to heat convection in fluid.
First order differential equation has a lot of application in the area of heat transfer which is discovered by Karthikeyan and Srinivasan Hassan and Zakari (2018). Also, the first order differential equation has many applications in temperature problem basically the ordinary differential equation. From which Hassan and Zakari Karthikeyan and Jayaraja (2016) found the use of newton’s law of cooling in solving some practical problem of first order ordinary differential equations. Mahanta et al. (2015, 2016) carried the analysis convective boundary condition. Rehan Hsu (2018) studied the first order differential equations and newtons law of cooling. Keryzig Rehan (2020), Caronongan Hsu (n.d.), Michael Keryszig (2006). carried the solution of first order and applications of differential equations has been considerd.
2. TYPES OF
DIFFERENTIAL EQUATION
· Ordinary
differential equations
The equations where the derivatives are taken with respect to only one variable is known as ordinary differential equations.
For e. g. dy/dx = sinx
· Partial
differential equations
The equation in which one variable depends on more than one variable is known as partial differential equations.
For e. g.
· Linear
differential equations
The linear differential equation is of the form dy/dx+p(x)y=Q(x)
· Nonlinear
differential equations
A nonlinear differential equation that is not a linear equation in the unknown function and its derivatives.
It is of the form
· Homogeneous
and nonhomogeneous differential equations
A homogeneous equation does have zero on the right side of the equality while a nonhomogeneous equations have a function on right side of the equations.
For e. g. (homogeneous differential equation)
(nonhomogeneous differential equation)
Now our main focus
is on solutions of first order differential equation and its application
3. Solution of first order differential equations
1) Using
integrating factor
The
linear differential equation is of the form
Then the integrating factor is defined by
the formula
IF =
Then the general solution is
Y =
2) Method
of variation of a constant
This method is similar
to integrating factor method. finding the general solution of the
homogeneous equation is the first necessary step.
The homogeneous equation is
The general solution of the homogeneous
equation always contains a constant C. The value of C we get after substituting
the solution into the nonhomogeneous differential equation. This method is
known as method of variation of a constant.
NOTE
The solution in both of above method is
always same.
Problem 1
Solve the differential equation
Solution
The given equation is already in a standard
form,
Therefore p(x)= 2x and q(x)= x
Now IF=
Now Y(x)= =
· Application of
firstorder differential equation to heat convection in fluid
1)
Heat transferring
Heat transferring is a process of
transfer of heat from a body with higher temperature to a body with lower
temperature. Hear the difference between the temperature is called potential
for which transfer of heat is happen. There is different mode for heat
transferring which are as follow
· Conduction
· Convection
· Radiation
1) Conduction
The process by which the heat is transfer
from hot end of an object to its cold end is called conduction. It is also
known as thermal conduction or heat conduction. Basically, in solid heat is
transferred by the process of conduction.
Figure 1
2) Convection
The process by which fluid molecules
moves from higher temperature region to lower temperature region is called
convection.
Figure 2
Figure 2 
3) Radiation
Radiation is the transfer of energy with
the help of electromagnetic wave. It is generated by the emission of
electromagnetic wave.
Figure 3
Figure 3 
So above we have seen that heat flowing
in solid by the process of conduction which we can determined by Fourier law.
And we see in fluid, heat flowing by convection which we can determine by Newton’s law of cooling.
NEWTON’S LAW OF COOLING
Statement
Let Q= heat absorbed
T =temperature of the body
=surrounding
temperature
Newton’s law of cooling state that the
temperature of the body and its surrounding is directly proportional to the
rate of cooling of the body.
i.e. (1)
or dQ = h [
Now integrating on both sides, we get
Hear , so heat
transform from
Again, for a
body of mass ‘M’, specific heat ‘’, temperature of the body ‘T’ kept in surrounding temperature ‘’
Then, heat =
mass*specific heat*body temperature
I. e. Q = M
Now rate of
colling is given by
Hence, we found
that
Due to
specific heat and mass of the body treated as constant so,
Hence the
above explanation suggested that as the time increases the difference between
the body temperature and surrounding temperature increases so that the rate of
temperature decreases.
Problem 2
A body cools
from 75 to 55 in 10
minutes when the surrounding temperature is 31. At what average temperature will its rate of
cooling be ¼ to that at the start.
Solution
Let be the
temperature of the surrounding
Consider a colling from
Initial
temperature = =
Final
temperature =
Time taken t =
10 min
We know
Consider the
rate of cooling when the temperature was
Rate of
cooling = ¼
Now
Hence at
temperature 39.5 the rate of
cooling be ¼th that at the start.
Problem 3
A heated metal ball is placed in cooler surroundings. Its rate of
cooling is 2 per minute
when its temperature is 60 and 1.2 per minute
when its temperature is 52. Find the temperature of the surroundings and the
rate of cooling when the temperature of the ball is 48.
Solution
Let b e the
temperature of the surrounding
Consider a
cooling at 60:
Temperature = 60
Rate of cooling = 2 per minute
By newton’s law of cooling,
(1)
Consider a
cooling at 52:
Temperature =52
Rate of cooling = 1.2 per minute
Again, by
newton’s law of cooling,
(2)
Now dividing
equation (1) by (2) we get,
Substituting
in equation (2) we get,
1.2 = h (5240)
To find rate
of cooling at 48:
By newton’s law of cooling,
Hence
temperature of the surrounding is 40 and rate of
cooling at 48 is 0.8 per minute.
Application of
first order differential equation: Orthogonal trajectory
Before going
to know what, orthogonal trajectory is, we have to
know trajectory first.
·
Trajectory
A curve which cut
every member of a given family of curve is called trajectory.
Figure 4
Figure 4 
·
Orthogonal
trajectory
A curve which
cuts every member of a family of curve at right angle is called orthogonal
trajectory. Or in other words it is the family of curve that intersect
perpendicularly to another family of curve.
Mathematically,
If we have a
family of curve given by F(x,y,a)
= 0 and another family of curve G(x,y,b). then the tangent of the curve is perpendicular to
each other. Where a and b are arbitrary constant.
Orthogonal trajectories are used in
mathematics for example as curve coordinate system or appear in physics as
electric field and its equipotential curve.
Step to find orthogonal
trajectory:
Case
1 for cartesian curve
1)
Differentiate
the given equation of family of curve and eliminate the parameter
2)
Replace
3)
Solve the new
differential equation and get orthogonal trajectory.
Case
– 2 for polar curve
1)
Differentiate
the given equation of family of curve with respect to .
2)
Replace
3)
Solve new
differential equation and get required solution
Problem 4
Find the
orthogonal trajectory of .
Solution
Given curve,
Differentiate the given equation w.r.t
x
Integrating on
both the side
Hence, the
orthogonal trajectory of
Problem 5
Find the
orthogonal trajectory of family of parabola
Solution
Given
Differentiate
both the side w.r.t x
Putting the value of a in the given equation
of curve we get,
(1)
This equation
is same as equation (1), so if we integrate this, we get the given curve
Hence the
orthogonal trajectory of is itself.
Problem 6
Find the
orthogonal trajectory of cardio’s
Solution
Given
Differentiate
both the side w.r.t
Eliminate a
from given equation
Integrating on
both side
, where b =
c/2
Hence r = b (1+cos
θ) is the orthogonal trajectory of r = a(1cos
θ)
4. Conclusion
In this paper
we discussed applications of first order differential equation that is newtons
law of cooling and orthogonal trajectory. Newton’s law of cooling states that
the rate at which an objects cools is directly proportional to the difference
in temperature between the object and its surrounding. It explains how fast an
object is cool down. However, it works only if the difference in temperature
between body and its surrounding must be small, the loss of heat from the body
should be by radiation only. And the major limitation of newtons law of cooling
is that the temperature of the surrounding must remain constant during the
cooling of the body.
Whereas
orthogonal trajectory is the tangent of two curve which are perpendicular to
each other. Here we see the use of first
order differential equation which allow these variables to be expressed
dynamically as a differential equation for the unknown position of the body as
a function of time. It has a major role in forensic science. The are many such
application of first order differential equation such as population growth and
decay, drug distribution in human body, survivability with AIDS, radioactive
decay and carbon dating, economics, and finance etc. finally this paper
believed that many problems of science and technology in future can be solve by
using first order differential equation.
CONFLICT OF INTERESTS
None.
ACKNOWLEDGMENTS
None.
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