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.

- is differentiable (as many times as the order of the equation) on and
- satisfies the differential equation for all . That is, for all .

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.

- Consider the differential equation
on
. We see that if we take
, then
is differentiable,
and therefore
- It can be easily verified that for
any constant
is a solution of the differential equation
- Consider the differential equation on . It can be easily verified that a solution of this differential equation satisfies the relation .

and move to higher orders later.

- Determine a differential equation for which a family of circles
with center at
and arbitrary radius,
is an implicit solution.
**Solution:**This family is represented by the implicit relation

where is a real constant. Then is a solution of the differential equation

The function satisfying (7.1.3) is a one parameter family of solutions or a general solution of (7.1.4). - Consider the one parameter family of circles
with center at
and unit radius. The family is represented by
the implicit relation
(7.1.5)

where is a real constant. Show that satisfies**Solution:**We note that, differentiation of the given equation, leads to

In Example 7.1.10.1, 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 7.1.5.2.

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

- Find the order of the following differential equations:
- Show that for each is a solution of
- Find a differential equation satisfied by the given family of curves:
- real (family of lines).
- real (family of parabolas).
- is a parameter of the curve and is a real number (family of circles in parametric representation).

- Find the equation of the curve which passes through and whose slope at each point is

A K Lal 2007-09-12