There are many branches of science and engineering where differential equations arise naturally. Now a days, it finds applications in many areas including medicine, economics and social sciences. In this context, the study of differential equations assumes importance. In addition, in the study of differential equations, we also see the applications of many tools from analysis and linear algebra. Without spending more time on motivation, (which will be clear as we go along) let us start with the following notations. Let be an independent variable and let be a dependent variable of . The derivatives of (with respect to ) are denoted by
The independent variable will be defined for an interval where is either or an interval With these notations, we are ready to define the term ``differential equation".
A differential equation is a relationship between the independent variable and the unknown dependent function along with its derivatives. More precisely, we have the following definition.
Some examples of differential equations are
In Example 7.1, the order of Equations 1, 3, 4, 5 are one, that of Equations 2, 6 and 8 are two and the Equation 7 has order three.
If is a solution of an ODE (7.1.1) on , we also say that satisfies (7.1.1). Sometimes a solution is also called an INTEGRAL.
Hence, is a solution of the given differential equation for all .
on any interval that does not contain the point as the function is not defined at . Furthere it can be shown that is the only solution for this equation whenever the interval contains the point .
is called a one parameter family of functions and is called a parameter. In other words, a general solution of (7.1.2) is nothing but a one parameter family of solutions of (7.1.2).
Now, eliminating from the two equations, we get
In Example 22.214.171.124, we see that is not defined explicitly as a function of but implicitly defined by (7.1.3). On the other hand is an explicit solution in Example 126.96.36.199.
Let us now look at some geometrical interpretations of the differential Equation (7.1.2). The Equation (7.1.2) is a relation between and the slope of the function at the point For instance, let us find the equation of the curve passing through and whose slope at each point is If is the required curve, then satisfies
It is easy to verify that satisfies the equation
A K Lal 2007-09-12