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

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

Rate Laws

An integrated rate law is an expression involving the rate of reaction and the species involved in the reaction

Consider the reaction:

A

The rate of change of A is dependent upon the concentration of A

First Order

First Order since it only depends upon the concentration linearly

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**Real life Application of Differential Equation**

They are used to model real life situations

Uses?

Physics – Schrodinger, atomic physics

Chemistry – Rate Laws, Statistical Thermo

…Synthetic Biology

dn

dt

=

-Kn

B

Second order

A Second Order rate law would depend upon

the square of the concentration:

Relationship between spices and total size of population

A useful parameter implicit in every pair of schedules of births and deaths is the intrinsic rate of natural increase, sometimes called the Malthusian parameter. Rates of reproduction and death rates evolve in concert -- when either is high the other is usually low.

A population whose size increases linearly in time would have a constant population growth rate given by

Growth rate of population = (Nt -N0) / (t -t0) = dN/dt = constant

Exponential population growth is described by the simple differential equation

Logistic Growth Model

Real-life populations do not increase forever. There is some limiting factor such as food or living space.

There is a maximum population, or carrying capacity, M.

A more realistic model is the logistic growth model where growth rate is proportional to both the size of the population (y) and the amount by which y falls short of the maximal size (M-y). Then we have the equation:

Some Application of Differential Equation in Engineering

Chemical reaction by Economists

A population of living creatures normally increases at a rate that is proportional to the current level of the population.

The electric current move through conductor by the potential difference that accrue in circuit

**Done By :**

Maryam Abdulnoor

Nooria Ibrahim

Shifaa marouf

Bahara Masoud

Maryam Abdulnoor

Nooria Ibrahim

Shifaa marouf

Bahara Masoud

REFERENCE

Andrewartha, H. G., and L. C. Birch. 1954. The distribution and abundance of animals. University of Chicago Press.

Davidson, J., and H. G. Andrewartha. 1948. Annual trends in a natural population of Thrips imaginis (Thysanoptera). J. Anim. Ecol. 17: 193-222.

Wikipedia

www.math/circle.com