THE APPLICATION OF FIRST ORDER DIFFERENTIAL EQUATIONS
VINOD Y
Lecturer of Mathematics
Govt. First Grade College, Tumakuru, Karnataka, India- 572102
ABSTRACT
The application of first order differential equation in temperature have been studied the method of separation of variables. Newton’s law of cooling were used to find the solution of the temperature problems that requires the use of first order differential equation and these solutions are very useful in Mathematics, Biology and Physics especially in analyzing problems involving temperature which requires the use of Newton’s law of cooling.
KEYWORDS: Differential equation and its application, Newton’s law of cooling.
INTRODUCTION
A first order differential equations is an equation that contain only first order derivative, and it has many application in mathematics, physics, engineering and many other subjects.
In some of the applications that are in mathematics, a first order differential equation plays a vital role in physics that includes a temperature problem which requires the use of Newton’s law of cooling of a particular substance.
APPLICATION OF FIRST ORDER DIFFERENTIAL EQUATION IN TEMPRATURE PROBLEMS
Newton’s law of cooling, which is equally applicable to heating, states that “ The rate of change of temperature of the body is proportional to the difference in temperature of the body and the medium in which it is placed ”.
i.e mathematically,
.............................. 
Where, T = Temperature of the body
S = Temperature of surroundings
K = Proportionality constant
The rate of change of temperature with respect to time
Here negative sign indicates “The decrease in temperature ”.
Clearly equation is first order differential equation, using variable separable method, let us find out the solution.
Consider 
Applying integration on both sides we get,
..............
Here C = Integration constant
To find ‘C’
Let 
Substitute value of C in
This is the required solution of Newton’s law of cooling.
Different forms of this equation are:-
To find proportionality constant, consider time interval
Where 
How Newton’s law of cooling serves reality let us see in terms of examples
Example: Suppose that a corpse(dead body) was discovered in a motel room at midnight and its temperature was 
, the temperature of the room is 
. Two hours later the temperature of the dead body dropped to 
find the time of death?
Given:= S =
= 0.1438
To find the time of death
here 
is normal temperature of the body before death
Now convert 
into 
The dead body was found at midnight so 
Now the difference of 
Death happened around 
APPLICATIONS
LIMITATIONS
CONCLUSION
We have seen that the application of first order differential equation in temperature problems are useful in mathematics and physics for instance in analysing problems involving temperature problems which requires the use of Newton’s law of cooling. when dealing with temperature problem it is recommended to use Newton’s law of cooling.
ACKNOWLEDGEMENT: I would like to express my gratitude towards to all my beloved teachers for their encouragement. I would like thank to Department of Mathematics GFGC Tumakuru for giving opportunity. Finally, I’m very grateful for support from my lovely parents and friends.
REFERENCES