DIFFERENTIAL EQUATIONS APPLICATIONS IN SCIENCE & ENGINEERING

Y.SARATH BABU A.SHAMEER AHMED

PRESENTED BY:

Contents

DefinationTypesProperties andApplications

DIFFERENTIAL EQUATIONS APPLICATIONS IN SCIENCE & ENGINEERING

Definition

These are the equations obtained eliminating of arbitrary constants from f(x,y,z,a,b)=0 equation in which a,b are constants.

A differential equation is an equation involving derivatives of an unknown function and possibly the function itself as well as the independent variable.

OR

Example

4 2 2 3sin , ' 2 0, 0y x y y xy x y y x 1st order equations 2nd order equation

Differential equations was

invented by LEIBNITZ

It was developed by JOHANN BERNOULLI

ORDER OF DIFFERENTIAL EQUATION The order of the differential equation is

order of the highest derivative in the differential equation.

Differential Equation ORDER

32 xdx

dy

0932

2

ydx

dy

dx

yd

364

3

3

ydx

dy

dx

yd

1

2

3

DEGREE OF DIFFERENTIAL EQUATION

Differential Equation Degree

032

2

aydx

dy

dx

yd

364

3

3

ydx

dy

dx

yd

0353

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.

Derivatives These Are Two Types

1. An ordinary differential equations

2. A partial differential equations

032

2

aydx

dy

dx

yd

32 xdx

dy

02

2

2

2

y

u

x

u

04

4

4

4

t

u

x

u

1

1

2

2

APPLICATIONS

Newton’s law of cooling

sTTdt

dT

Ex: A murder victim is discovered and a lieutenant was to estimate the time of death. The body is loacted in a room that body kept at a constant

temperture of 68◦F . The lieutenant arrived at 9.30P.M and measured the body temperture as 94.4◦F at that time. Another measurement of

the body temperture at 11P.M is 89.2◦F

Ans : time of death 53.8 minutes

Rate Of Decay Of Radioactive Materials

y is the quantity present at any time(t)

dy

ydt

─

Newton's Second Law In Dynamics

Law of natural growth or decay

N(t) is amount of substance at ‘t’

In Schrodinger

Wave Equation

The Schrodinger equation is the name of the basic non-relativistic wave equation used in one version of quantum mechanics to describe the behaviour of a particle in a field of force. There is the time dependant equation used for describing progressive waves, applicable to the motion of free particles. And the time independent form of this equation used for describing standing waves.

In Laplace transforms

RL circuit

L di/dt + Ri = E

Heat Equation In Thermo Dynamics

Example:

10

PHYSICAL ORIGIN1. Free falling stone g

dt

sd

2

2

2. Spring vertical displacement ky

dt

ydm

2

2

where y is displacement, m is mass and

k is spring constant

a=-g

JacobianProperties•If the Jacobian(J) value is zero then the given two relations are dependent.•If the Jacobian(J) value is not zero then the given two relations are independent.