Singular Solutions of First Order ODEs - 1 | Mathematics for IIT JAM, GATE, CSIR NET, UGC NET PDF Download

1 What is a differential equation?

Discussion: A differential equation is an equation involving derivatives.

EXAMPLES:

In the first example, y is to be a function of x whose derivative is 2x² plus 3 times the original function y. Note that we write y instead of y(x). We will usually not distinguish between functions and variables in our notation. It will actually make things easier in some problems to just think of functions as variables and treat them accordingly. We will use both the notations dy/dx and y' (x) to express the derivative of y with respect to x in this class, though we will often abbreviate y' (x) as just y' , just as we write y instead of y(x).

The order of a differential equation is the highest order of derivative that occurs in the equation. The first and third examples are first order while the second example is second order. In this first chapter, we will consider first order equations. Selected equations of higher order will be considered in subsequent chapters.

Consider the first order equation y' = x − y. We can interpret this equation both algebraically and geometrically. Algebraically, it means we are looking for a function, y(x), whose derivative is equal to the variable x minus the function y(x) itself. To interpret the equation geometrically, we recall that the derivative y' is the slope of a curve. So this equation means we are looking for a curve whose slope at the point (x, y) is x − y. It is easiest to understand what this geometric condition means using a type of graph called a slope field. We draw a small line segment at each point (x, y) with slope f (x, y). These form a “slope” field. Figure 1 shows the slope field for the equation dy/dx = x − y.

Of course, we can’t actually draw a line segment at every point (x, y), that would be infinitely many line segments. Indeed, it’s a lot of work to just draw line segments for a 13 × 13 grid. But that sort of tiresome repetitive task is exactly what computers were made for. A slope field is similar to a vector field, which you studied in Calculus 3, except that in a vector field we plot vectors which have both a length and a direction, while in a slope field we are just concerned with slope, that is direction, and not with length.

A particular solution to a differential equation is a function that satisfies the equation.

Example:

y = x − 1 is a particular solution to y' = x − y.

Algebraically, it is easy to check that y(x) = x − 1 is a solution to y' = x − y. Just compute y' = 1 = x − (x − 1) = x − y. Geometrically, the solution curve, y = x − 1 must run tangent to the slope field since at every point the solution curve has the same slope as the slope field, as illustrated in figure 2. A curve that runs tangent to the slope field at every point is called an integral curve of the slope field (or an integral curve of the differential equation).

The reason for emphasizing that y = x − 1 is a particular solution, rather than just calling it a solution, to y' = x − y, is that the differential equation actually has lots of different solutions and we want to emphasize that y = x − 1 is just one particular solution out of infinitely many possibilities.

Examples: 

y = e^−x + x − 1 is a particular solution to y' = x − y

y = 2e^−x + x − 1 is a particular solution to y' = x − y

Looking at the slope field, it is easy to see that we could draw infinitely many different curves along the arrows, each of which corresponds to a solution of the equation (see figure 3).

There are two common approaches to dealing with the fact that we have infinitely many different solutions. One is to ask for the general solution instead of the particular solution.

A general solution is a set of solutions to a differential equation with as many arbitrary constants as the order of the equation.

EXAMPLE:

y = ke^−x + x − 1 is the general solution to y' = x − y.

For any choice of the arbitrary constant k we get a solution to the equation. The three solutions above come from choosing k = 0, k = 1 and k = 2. The equation is first order so we only expect one arbitrary constant. In many cases the general solution gives all the solutions but this need not be the case. A singular solution is a particular solution to an equation which is not an instance of the general solution. We will see examples of this later. (The term singular solution is sometimes given a more restrictive and technical meaning in advanced courses.)

A second approach to dealing with the non-uniqueness of solutions is to narrow the problem. We ask not for any old solution to the problem but rather for a solution which also has some additional properties. One common situation is an initial value problem where one wants the solution to a differential equation which takes a specified value at a given point.

EXAMPLE: The unique solution to the initial value problem y' = x − y, y(−3) = 0 is y = (4e³)ex + x − 1.

The condition y(−3) = 0 means that when x = −3, y = 0. Geometrically, this means that we are looking for a curve that runs along the slope field, passing through the point (x, y) = (−3, 0) (see figure 4). It should be clear from the picture that the solution is unique, that is to say, there is only one solution to the initial value problem. Once we have picked a point on the figure, then the curve must move along the arrows so there is only one way it can go (see figure 4).

Let’s finish this introduction with the question of why one might want to solve differential equations. Differential equations arise in a great many applications. Consider Newton’s second law of motion

where p is momentum and t is time. This is probably more familiar to you as F = ma, where we have made the assumption that mass is constant. Suppose you are told the acceleration of gravity on the moon is 1.6m/sec² . If you drop a mass above the surface of the moon (“drop” implying released gently with initial velocity 0), how far will it fall in 1 second? Here you know acceleration, but what you want to know is distance. If we let x(t) be the distance fallen after time t seconds, then the acceleration is the second derivative of the distance, so x'' = 1.6. Furthermore, when t = 0 the mass has not yet fallen any distance, so x(0) = 0 and since the velocity at time t = 0 is 0, we also have x' (0) = 0. Thus we must solve the second order initial value problem, x'' = 1.6, x(0) = 0, x' (0) = 0. Since force is defined using derivatives, in any physical situation where you know the forces acting on an ob ject and then have to determine the future behavior of the ob ject you will have a differential equation. Of course differential equations occur in many other sub jects besides physics. Many situations can be pictured as slope fields, where you can tell how things are changing in the short run given knowledge about where you are right now. Working out how things change over the long run is then finding an integral curve for the slope field, or equivalently, solving a differential equation.