2.  THE TERM DIFFERENTIAL EQUATION
 ORDINARY DIFFERENTIAL EQUATION
PARTIAL DIFFERENTIAL EQUATION
ORIGIN $ APPLICATION OF DIFFERENTIAL
EQUATION

3. ORIGIN OF
DIFFERENTIAL EQUATION
Differential equations first came into existence with the
invention of calculus by Newton and Leibniz. In Chapter 2 of
his 1671 work “ Methodus fluxionum et Serierum
Infinitarum ",[1] Isaac Newton listed three kinds of
differential equations:
He solves these examples and others using infinite series and
discusses the non-uniqueness of solutions.

4. Differential equation
A differential equation contains
one or more terms involving
derivatives of one variable (the
dependent variable, y) with
respect to another variable (the
independent variable, x).

5. A DIFFERENTIAL EQUATION INVOLVING ORDINARY DERIVATIVES OF
ONE OR MORE DEPENDENT VARIABLE WITH RESPECT TO A SINGLE
INDEPENDENT VARIABLE IS CALLED DIFFERENTIAL EQUATION.

6. y is dependent variable and x is independent variable,
and these are ordinary differential equations
32  x
dx
dy 032
2
 ay
dx
dy
dx
yd
36
4
3
3






 y
dx
dy
dx
yd

7. THE ORDINARY DIFFERENTIAL
EQUATION IS FURTHER
CLASSIFIED AS;
LINEAR ORDINARY
DIFFERENTIAL EQUATION
NON-LINEAR ORDINARY
DIFFERENTIAL EQUATION

8. EXPONENTIAL GROWTH-POPULATION
NEWTON’S LAW OF COOLING
EXPONENTIAL DECAY-RADIOACTIVE MATERIAL
FALLING OBJECT
R-L CIRCUIT

11. In 1980 the population in lane country was 250,000.this grew to 280,000 in
1990

12. The percent change from one
period to another is calculated from
the formula:
Where:
PR = Percent Rate
VPresent = Present or Future Value
VPast = Past or Present Value
The annual percentage growth rate is
simply the percent growth divided by N, the
number of years.

13. Newton's Law of Cooling states
that the rate of change of the
temperature of an object is
proportional to the difference
between its own temperature
and the ambient temperature
(i.e. the temperature of its
surroundings).

14. Newton's Law makes a statement about
an instantaneous rate of change of the
temperature. We will see that when we
translate this verbal statement into a
differential equation, we arrive at a
differential equation. The solution to this
equation will then be a function that
tracks the complete record of the
temperature over time. Newton's Law
would enable us to solve the following
problem.

15. Introduction to
differential
equation

16. Order and degree

17. Order of Differential Equation
The order of the differential equation is order of the highest
derivative in the differential equation.
Differential Equation ORDER
32  x
dx
dy
0932
2
 y
dx
dy
dx
yd
36
4
3
3






 y
dx
dy
dx
yd
1
2
3

18. Degree of Differential Equation
Differential Equation Degree
032
2
 ay
dx
dy
dx
yd
36
4
3
3






 y
dx
dy
dx
yd
03
53
2
2












dx
dy
dx
yd
1
1
3
The degree of a differential equation is power of the highest
order derivative term in the differential equation.

19. General and particular solutions
GENERAL SOLUTION;(COMPLETE SOLUTION)
A SOLUTION OF D.E IN WHICH THE NUMBER OF
ARBITRARY CONSTANT IS EQUAL TO THE ORDER OF D.E
PARTICULAR SOLUTION;
a form of the solution of a differential equation with
specific values assigned to the arbitrary constants.

20. WRONSKIAN THEOREM
Let f1, f2,...,fn be functions in C[0,1] each of which
has first n-1 derivatives. If the “ Wronskian ” of
this set of functions is not identically zero then
the set of functions is linearly independent.

21. W(Y1,Y2)=
Y1 Y2
Y1’ Y2’
IF W(Y1,Y2)= 0
THEN Y1 AND Y2 ARE CALLED LINEAR
INDEPENDENT

22. HOMOGENOUS AND NOBN HOMOGENOUS DIFFERENTIAL
EQUATION
HOMOGENEOUS LINEAR ORDINARY
DIFFERENTIAL EQUATION OF SUPER
POSITION OFSOLUTION WILL CONTAIN
ONLY
Y=C.F

23. IN NON HOMOGENEOUS
DTFFERENTIAL EQUATION
THE SOLUTION CONTAIN ,
Y=C.F+P.I

24. SUPER POSITION PRINCIPLE
IT STATES THAT,WE CAN
OBTAIN FURTHER
SOLUTIONS FROM GIVEN
ONES BY ADDING THEM
OR BY MULTIPLY THEM
WITH ANY CONSTANT

25. PHYSICAL
APPLICATIONS OF
DIFFERENTIAL EQUATION

26. •SIMPLE HARMONICMOTION
•SIMPLE PENDULUM
•OSCILLATIONSOF A SPRING
•OSCILLATORY ELECTRICALCIRCUIT
•ELECTRO-MECHANICALANALOGY
•DEFLECTIONOF BEAMS
•WHIRLING OF SHAFTS
•HOOKE’SLAW

27. Whirling speed is also called as Critical speed of
a shaft. It is defined as the speed at which a
rotating shaft will tend to vibrate violently in the
transverse direction if the shaft rotates in
horizontal direction. In other words, the
whirling or critical speed is the speed at which
resonance occurs.

28. Thank you….