Motivation and Early Developments Just we look at some real life situations where ● the motion of projectile, rocket, satellite and planets ● the charge or current in the electric circuit ● the conduction of heat on a rod or in a slab ● the vibrations of a wire or membrane etc are to be determined. The mathematical formulations of such problems emerge as differential equations under certain scientific laws. These laws involve various rates of change (derivatives) of one or more quantities with respect to other quantities. Thus the scientific laws manifest as mathematical equations involving derivatives, viz.
differential equations. Differential Equations emanate from the problems in geometry, mechanics, physics, chemistry, and engineering studies. We have studied about “rates” in our early classes. This is also known as instantaneous rate of change which is denoted as dy dx .
We give below some relations between the rate of change and unknown functions that occur in real life situations. (a) The rate of change of y with respect to x is directly proportional to y : dx = ky . (b) The rate of change of y with respect to x is directly proportional to the product of y and x : dx = ky x . (c) The rate of change of y with respect to x is inversely proportional to y : dx = k y .
(d) The rate of change of y with respect to x is directly proportional to y and inversely proportional to x : dx = k y A differential equation is an equation in which some derivatives of the unknown function occur. In many cases the independent variable is taken to be time. - - Ordinary Differential Equations In order to apply mathematical methods to a physical or “real life” problem, we must formulate the problem in mathematical terms; that is, we must construct a mathematical model for the problem. Many physical problems concern relationships between changing quantities.
Since rates of change are represented mathematically by derivatives, mathematical models often involve equations relating to an unknown function and one or more of its derivatives. Such equations are differential equations. They are of basic significance in science and engineering since many physical laws as well as relations are modelled in the form of differential equations. Differential equations are much useful in describing mathematical models involving population growth or radio-active decay.
The study of biological sciences and economics is incomplete without the application of differential equations. The subject of di ff erential equations was invented along with calculus by Newton and Leibniz in order to solve problems in geometry and physics. It played a crucial part in the development of Newtonian physics by the Bernoulli family, Euler, and others. Some of the applications of differential equations in our daily life are found in mobile phones, motor cars, air flights, weather forecast, internet, health care, or in many other daily activities.
In this chapter, we introduce and discuss the first order ordinary differential equations and some methods to find their solutions.