INTRODUCTION TO DIFFERENTIAL

EQUATION

Honey JoseEC.BNO.1

THE TERM DIFFERENTIAL EQUATION

ORDINARY DIFFERENTIAL EQUATIONPARTIAL DIFFERENTIAL EQUATION

ORIGIN $ APPLICATION OF DIFFERENTIAL EQUATION

THE MAIN TOPICS INCLUDED:

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.

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).

ORDINARY DIFFERENTIAL EQUATION

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

DIFFERENTIAL EQUATION.

y is dependent variable and x is independent variable, and these are ordinary differential equations

32 xdxdy 032

2

aydxdy

dxyd

364

3

3

ydxdy

dxyd

THE ORDINARY DIFFERENTIAL EQUATION IS FURTHER

CLASSIFIED AS;

LINEAR ORDINARY DIFFERENTIAL EQUATION

NON-LINEAR ORDINARY DIFFERENTIAL EQUATION

APPLICATIONS OF DIFFERENTIAL EQUATION

EXPONENTIAL GROWTH-POPULATIONNEWTON’S LAW OF COOLING

EXPONENTIAL DECAY-RADIOACTIVE MATERIAL

FALLING OBJECTR-L CIRCUIT

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

The percent change from one period to another is calculated from the formula:  

Where:PR = Percent RateVPresent = Present or Future ValueVPast = Past or Present ValueThe annual percentage growth rate is simply the percent growth divided by N, the number of years.

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).

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.

Introduction to

differential equation

Order and degree

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

dxdy

0932

2

ydxdy

dxyd

364

3

3

ydxdy

dxyd

1

2

3

Degree of Differential Equation

Differential Equation Degree

032

2

aydxdy

dxyd

364

3

3

ydxdy

dxyd

0353

2

2

dxdy

dxyd

1

1

3

The degree of a differential equation is power of the highest order derivative term in the differential equation.

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.

WRONSKIAN THEOREMLet 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. 

W(Y1,Y2)= Y1 Y2Y1’ Y2’

IF W(Y1,Y2)= 0

THEN Y1 AND Y2 ARE CALLED LINEAR INDEPENDENT

HOMOGENOUS AND NOBN HOMOGENOUS DIFFERENTIAL

EQUATION

HOMOGENEOUS LINEAR ORDINARY DIFFERENTIAL EQUATION OF SUPER

POSITION OFSOLUTION WILL CONTAIN ONLYY=C.F

IN NON HOMOGENEOUS DTFFERENTIAL EQUATION THE SOLUTION CONTAIN ,

Y=C.F+P.I

SUPER POSITION PRINCIPLE

IT STATES THAT,WE CAN OBTAIN FURTHER

SOLUTIONS FROM GIVEN ONES BY

ADDING THEM OR BY MULTIPLY THEM WITH

ANY CONSTANT

PHYSICAL APPLICATIONS OF

DIFFERENTIAL EQUATION

• SIMPLE HARMONIC MOTION

• SIMPLE PENDULUM

• OSCILLATIONS OF A SPRING

• OSCILLATORY ELECTRICAL CIRCUIT

• ELECTRO-MECHANICAL ANALOGY

• DEFLECTION OF BEAMS

• WHIRLING OF SHAFTS• HOOKE’S LAW

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.

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