First Order Ordinary
Differential Equations
Simply put, a differential equation is an equation that contains derivatives. To solve an equation of this type is to find a function that satisfies the differential equation. The difference between solving an algebraic equation and solving a differential equation is that with the former you are looking for a number and with the later you are looking for a function.
Before solving a differential equation, it is useful to determine how it is classified. The trick to solving these equations is to recognize the Type, Order, Degree, and Linearity of the equation. Once determined, obtaining a solution is a matter of using time tested methods.
The student may be confused as to the terminology used in some cases because of how some names conflict with how they are used in past studies. For example, the term linearity means something different when discussing differential equations than when discussing algebraic equations. This confusion is compounded by some of the notation used in LiveMath These inconsistencies will be highlighted as the terminology is presented.
An Ordinary Differential Equation (ODE), is an equation that contains only ordinary derivatives with respect to a single independent variable. A Partial Differential Equation (PDE), is an equation that contains partial derivatives of two or more independent variables. Using customary notation, the first example below is an ODE, the second a PDE.
LiveMath's use of partial derivative notation for its derivative
Op may cause you to have difficulty here. In most texts, it is
customary to use the Greek letter 
to denote
partial derivatives, and therefore when LiveMath's partial derivative
Op is used to study Ordinary Differential Equations, you may become
confused. The best advise here is to realize that there is a difference.
Just remember that, since the subject is Ordinary Differential
Equations, the fact that the Partial Derivative Op is used for
some operations, does not necessarily mean the equation is a Partial
Differential Equation.
In LiveMath, you can construct an ordinary derivative Op by using the differential operator d. The first graphic below is an example. The second is a different style which you will use in the study of Separation of Variables.
Since LiveMath's Partial Derivative Op (third graphic above) does not take partial derivatives until an Independence Declaration is created, it is much easier, in most cases, for you to use this built-in Op rather than construct one.
The use of partial derivatives to test for exactness and their use to solve Linear Equations may exacerbate the situation. Again, it is best to remember that the subject is ODE's and that partial differential equations have more than one independent variable. The equations you will be working with only have one.
The order of a differential equation is the order of the highest appearing derivative in the equation. The following equation is, therefore, a second-order equation: note
The second order derivative above demonstrates the use of LiveMath's partial derivative Op to denote an ordinary differential equation. The same equation using conventional notation is shown below.
The problem with using the constructed OP above, in LiveMath, is that it will not work unless it is placed in the following form:
If you can write an ODE as a polynomial in the dependent variable and its derivatives, then its degree is the power to which the highest-order derivative is raised. The equation below is a first degree equation because the highest order derivative (the 4th) is to the first power.
A linear differential equation takes the following form:
There are two items to note in this equation. First, note that the derivatives and the dependent variable, y, are of the first degree. Second, the coefficients depend only on the independent variable (x, in this case). In the context of differential equations, a coefficient is a constant or it can be a function of the independent variable.
In addition, although not apparent from the equation above, the dependent variable is not the argument of a transcendental function. The independent variable can be an argument, but the dependent variable cannot.
The following examples are graphically displayed to help you learn to distinguish linear equations from non-linear equations:
In differential calculus you learn to take the derivative of a function. The resulting equation contains a new function along with a derivative. This resulting equation is a differential equation. The following is an example.
In this section, your goal is to reverse this process. Given a differential equation; what is the function that gives you this equation? You are looking for the solution to the differential equation!
Another way of putting it is: The function y = f(x), is said to be a solution to a differential equation, if when substituted into that differential equation, it reduces the equation to an identity. For example, the differential equation generated above is reduced to an identity when the original function (the solution) is substituted into it:
Therefore y = sin(x) is a solution to the differential equation dy/dx = cos(x).
A solution that you can place in the form: y = f(x), is called an explicit solution. The unknown (dependent) variable y, is a function of the known (independent) variable x. In the example above, y = sin(x) is an explicit solution of the differential equation dy/dx = cos(x)
When you solve an ODE and cannot isolate the dependent variable,
as you could in the case above, then you have an implicit solution.
You will normally place these solutions in the form; f (x,y)
= 0. Implicit solutions assume there is at least one explicit
solution in the interval in question. For example, the equation
for a circle is an implicit solution of the differential equation
dy/dx=-x/y with two explicit solutions; 
and
LiveMath is very useful in helping you determine whether a function is a solution to a given differential equation. Below is an exercise that demonstrates.
Expl #1 Testing Solutions Option/Right-Click to download
Expl# files in Mac and Win Netscape.
When you substitute the y-Props from inside the Case Theories into the ODE Prop, notice how LiveMath brings the differential equation into each Case Theory allowing you to test each solution separately.
You can also use one Case Theory to test several potential solutions by turning LiveMath's 'Always ReCompute' command on. Below, the example is displayed using this method:
The result after the solution is altered:
When you study integration you learn that a constant of integration is generated as part of the process. If a second integration is performed a second constant is generated and LiveMath keeps track of these by using numerical subscripts (ie; c_100, c_101) note.
When solving first-order differential equations for what is called the General Solution, you obtain a family of curves (functions) that contain one "arbitrary" and "essential" constant. It is called arbitrary, because it can be any constant, and it is called essential, because it cannot be replaced by a smaller number of constants (this concept is expanded upon, later). When you solve a second order differential equation for its general solution, you should expect to obtain two arbitrary and essential constants.
It is customary to call a solution which contains n constants an n-parameter family of solutions, where each constant is referred to as a parameter. Therefore, when you solve an nth-order differential equation you should expect, in most cases, an n-parameter family of solutions( in most cases that is, because there are a small number of equations where this may not the case). Since you will not be looking at higher order equations in this handbook, assume the former.
When you look at 1st Order equations, as will be the case in this e-book, you can assume that there is a set of solutions for the defined interval. When a solution contains one arbitrary constant, then the solution is said to be a General Solution.
Given an Initial Value, you can turn a General Solution into a single solution. This new equation, which now does not contain an arbitrary constant, but an actual one, is called a Particular Solution.
It is customary to assume that the General Solution of a differential Equation contains every Particular Solution. This is not the case for all equations, however, if, in fact, there is a solution (called a Singular Solution) which you cannot obtain by assigning a value to the constant in the General Solution. This subject will not be covered in this e-book.
First Order Ordinary
