APPLICATIONS OF DIFFERENTIAL EQUATIONS- ANIL. S. NAYAK

WHAT IS A DIFFERENTIAL EQUATION????

A differential equation is a mathematical equation that relates some function of one or more variables with its derivatives.

Differential Eqn. are broadly classified into :Ordinary and Partial Differential equations

HISTORY OFDIFFERENTIAL EQUATIONS

In mathematics, the history of differential equations traces the development of "differential equations" from calculus, which itself was independently invented by English physicist Isaac Newton and German mathematician Gottfried Leibniz. “Differential equations began with Leibniz, the Bernoulli brothers, and others from the 1680s, not long after Newton’s ‘fluxional equations’ in the 1670s.”

Example:A population grows at the rate of 5% per year. How long does it take for the population to double? Use differential equation for it.

Solution: Let the initial population be P0 and let the population after t years be P, then

dP 5 dP P dP 1= P = = dt

dt 100 dt 20 P 20

[Integrating both sides]

e1

log P = t +C20

dP 1= dt

P 20

Solution Cont.At t = 0, P = P0 e 0 e 0

1×0log P = + C C = log P

20

e e 0 e0

1 Plog P = t+log P t =20 log

20 P

0When P = 2P , then

0e e

0

2P 1t =20 log = log 2 years

P 20

Hence, the population is doubled in e20 log 2 years.

RADIOACTIVE HALF-LIFE

• A stochastic (random) process• The RATE of decay is dependent upon the

number of molecules/atoms that are there• Negative because the number is decreasing• K is the constant of proportionality

kNdt

dN

ATOMIC PHYSICS

TIRE MODELING

MECHANICAL VIBRATION ANALYSIS

EARTHQUAKE ANALYSIS