Ordinary Differential Equations -- by Understandable Primers GmbH

Other writeups on the subject and Banach spaces notwithstanding, this is a subject which the average Joe can understand. Although a thorough familiarity with calculus is assumed for the first course in ordinary differential equations, it is not necessary to follow what we are talking about here. Basic knowledge in college algebra will suffice to get it. Let's begin.

An ordinary differential equation is an equation involving a dependent variable (let's say y) and its derivative (y') with respect to some independent variable (let's say t, resulting in dy/dt.) A derivative is the rate of change of a particular quantity. The derivative of position, for example is velocity, and velocity's derivative is acceleration. When velocity and position are related, we have a differential equation. Imagine a road that gets bumpier the farther you drive on it. This means that your car will deccelerate as your position increases (i.e. as you drive further along the street.) This is a second order differential equation, because the second derivative of position, acceleration, is related to the position itself by the friction and bumpiness of the road.

Now there are some terms that we need to define.

Linear: A linear differential equation is one that is linear with respect to y, the dependent variable in question. This means that there are no odd powers of y or other strange functions having y as an argument. (No sin (y) or ln (4y), for example.) A linear equation can be written in the following form:

Qn(t)*y(n) + Qn-1(t)*y(n-1) + . . . + Q2(t)*y'' + Q1(t)*y' + Q0(t)*y = R(t)

Here the Q's represent generic functions of t which are coefficients of y and its derivatives, one function per y-term. R(t) is also a function of t which is not attached to some form of y. (We will talk more about R later.) Also, the y(n) means the nth derivative of y. So, in more understandable English, we have a series of y and any number of its derivatives, each with a coefficient which is a function of t but NOT of y. The fact that Q and R are only functions of t makes a function linear.

Homogenous: An equation is homogenous if there is no term which is not multiplied by the dependent variable or one of its derivatives. Did that make sense? Let's try again. Look at the general linear equation above. See that R-term? It has no y and no derivative of y attached to it. That makes our general equation nonhomogenous. If, and only if R is 0, does our equation have the general form below.

Qn(t)*y(n) + Qn-1(t)*y(n-1) + . . . + Q2(t)*y'' + Q1(t)*y' + Q0(t)*y = 0

Note especially that in this example, we again have Q(t), a function of t only. This makes our function again linear, which is normal for a basic course in differential equations. And since the R-term is 0, the function is still homogenous.

Autonomous: Autonomous differential equations have close to the same general form that we have been discussing, except that there is no direct expression of the independent variable t. In other words, nowhere in the equation does the letter t appear directly, even though y is hopefully a function of t for some values. This means that the equation looks like this:

Qn*y(n) + Qn-1*y(n-1) + . . . + Q2*y'' + Q1*y' + Q0*y = R

This equation looks suspiciously like our general form for a homogenous equation above, except now our Q-terms are not functions of t, as we have discussed, and the R-term is back, because this equation is not necessarily homogenous. Because these Q and R-terms are now constants, there are a variety of useful methods for solving them which would be covered in a typical course on ODE's, such as the characteristic equation, undetermined coefficients, and variation of parameters in combination with the Wronskian of the equations fundamental set. Does that sound interesting? It is! Take your nearest course in ordinary differential equations and you will learn all this and more.

There is, however, no requirement that an autonomous equation be linear, which gives a much less complicated general form, but one that is harder to wrap your brain around:

F(y,y',...,y(n)) = 0

We see here that the independent variable, t, is not present in the function F, which represents the whole differential equation. F is only a function of y and its derivatives, and can certainly be non-linear, which makes it much more difficult to solve.

That is a simple introduction to a few terms that apply to ordinary differential equations. While solution methods are outside the scope of this writeup, the aspiring reader, who may be "needing these wisdoms bad", is strongly encouraged to, if still attending a university or college, take a calculus refresher and dive in! Those who now have a job/wife/child are advised to take it slower and invest in an online textbook in ODE's, such as one of the following:

  • Ordinary Differential Equations for Reliability, Maintainability, and Availability
    Located at: http://mathforum.org/differential/watson/tex/
    As .tex files, to be compiled with any TeX capable editor
  • Difference Equations to Differential Equations
    Located at: http://math.furman.edu/~dcs/book/
    As .pdf or PostScript files

The author, having compiled this information based on personal experience/knowledge and with a bit of advice from Wolfram's Mathworld, wishes you the best of luck!