INTRODUCTION TO DIFFERENTIAL EQUATION Honey Jose EC.B NO.1
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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