4.1 Basics of differential equations  (Page 2/14)

It is convenient to define characteristics of differential equations that make it easier to talk about them and categorize them. The most basic characteristic of a differential equation is its order.

General and particular solutions

We already noted that the differential equation $y^{\prime }=2x$ has at least two solutions: $y=x^2$ and $y=x^2+4.$ The only difference between these two solutions is the last term, which is a constant. What if the last term is a different constant? Will this expression still be a solution to the differential equation? In fact, any function of the form $y=x^2+C,$ where $C$ represents any constant, is a solution as well. The reason is that the derivative of $x^2+C$ is $2x,$ regardless of the value of $C.$ It can be shown that any solution of this differential equation must be of the form $y=x^2+C.$ This is an example of a general solution to a differential equation. A graph of some of these solutions is given in [link] . ( Note : in this graph we used even integer values for $C$ ranging between $\mathrm{-4}$ and $4.$ In fact, there is no restriction on the value of $C;$ it can be an integer or not.)

In this example, we are free to choose any solution we wish; for example, $y=x^2-3$ is a member of the family of solutions to this differential equation. This is called a particular solution    to the differential equation. A particular solution can often be uniquely identified if we are given additional information about the problem.

Initial-value problems

Usually a given differential equation has an infinite number of solutions, so it is natural to ask which one we want to use. To choose one solution, more information is needed. Some specific information that can be useful is an initial value , which is an ordered pair that is used to find a particular solution.