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## Tuesday, 5 August 2014

### CHAPTER 1 - Differential Equations - Examples from the Real World

A differential equation can simply be said to be an equation involving derivatives of an unknown function. For example, consider the equation
 $\dfrac{{dy}}{{dx}} + xy = {x^2}$
This is a differential equation since it involves the derivative of the function $y(x)$ which we may wish to determine. We must first understand why and how differential equations arise and why we need them at all. In general, we can say that a differential equation describes the behaviour of some continuously varying quantity.                                 Scenario – 1: A freely falling body
A body is released at rest from a height $h$. How do we described the motion of this body ? The height $x$ of the body is a function of time. Since the acceleration of the body is $g$, we hav
 $\dfrac{{{d^2}x}}{{d{t^2}}} = - g$
This is the differential equation describing the motion of the body. Along with the initial condition $x(0) = h,$ it completely describes the motion of the body at all instants after the body starts falling.                                                                                                  Scenario – 2: Radioactive disintegration
Experimental evidence shows that the rate of decay of any radioactive substance is proportional to the amount of the substance present, i.e.,
 $\dfrac{{dm}}{{dt}} = - \lambda m$
where $m$ is the mass of the radioactive substance and is a function of $t$. If we know $m(0)$, the initial mass, we can use this differential equation to determine the mass of the substance remaining at any later time instant                                                                 Scenario – 3: Population growth
The growth of population ( of say, a biological culture) in a closed environment is dependent on the birth and death rates. The birth rate will contribute to increasing the population while the death rate will contribute to its decrease. It has been found that for low populations, the birth rate is the dominant influence in population growth and the growth rate is linearly dependent on the current population. For high populations, there is a competition among the population for the limited resources available, and thus the death rate becomes dominant. Also, the death rate shows a quadratic dependence on the current population.
Thus, if $N(t)$ represents the population at time $t$, the differential equation describing the population variation is of the form
 $\dfrac{{dN}}{{dt}} = {\lambda _1}N - {\lambda _2}{N^2}$
where ${\lambda _1}$ and ${\lambda _2}$ are constants.
Along with the initial population $N(0)$, this equation can tell us the population at any later time instant.