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12 March 2004

More About Integral Curves

by George E. Hrabovsky, President, MAST

What We Did Last Time

We explored integral curves and their relationship to differential equations.

An Additional Note

Some time ago one of you sent me a link to a nice differential equations site. I have not been able to find that email, so if you could send it to me again I would appreciate it.

Where We Will Go This Time

We will explore more of the properties of integral curves and introduce the idea of direction fields.

Direction Fields

Last time we introduced Picard's theorem that stated that within some closed region a differential equation has a solution at every single point so long as the function for the differential equation is continuous.  If we plot a short segment at each point in a region such that the slope of the segment represents the tangent of the solution at that point, then we will better understand how those solutions fit together.  Such a set of plots is called a direction field.  Let us make an example,

Given a rectangular region we see that the direction field is.

Each of these segments represents a possible solution to the differential equation.  We can see that the general solution can be found by first factoring,

and then separating the variables,

We then integrate from the initial values,

so,

If we choose then the solution will be,

and the integral curve can be plotted.

We can superimpose this onto the direction field.

Mathematics Challenge from Last Time

One thing we can do is find the critical values of the curve.  For details see, "Extreme Values of Functions," and "Extreme Values of Functions II."

Mathematics Challenge

What else can we learn about curves using derivatives?

Sources About Differential Equations

On-Line:

The ultimate resource for those interested in research related to differential equations is the Electronic Journal of Differential Equations, which is free:

http://ejde.math.swt.edu/

Another nice site that covers lots of topics, though with only a few examples, is the SOS Mathematics page for differential equations:

http://www.sosmath.com/diffeq/diffeq.html

Another free online journal is the Electronic Journal of Qualitative Theory of Differential Equations:

http://www.math.u-szeged.hu/ejqtde/

An online course in differential equations is given at this site:

http://math.stcc.edu/DiffEq/DiffEq.html

Books:

Martin Braun (1993), Differential Equations and Their Applications, Springer-Verlag.  This is a nice book crammed full of applications.

George F. Simmons (1991), Differential Equations with Applications and Historical Notes, McGraw-Hill.  This is one of my favorite differential equations books, it starts out at an elementary level and includes nonlinear equations and partial differential equations.  My columns will loosely follow this book up to a point and will then take a more geometrical viewpoint.

John W. Dettman (1974), Introduction to Linear Algebra and Differential Equations, McGraw-Hill (reprinted in 1986 by Dover Publications). This is a wonderful book that is also inexpensive; it begins with a discussion of complex numbers, then linear algebra, and then differential equations.

Morris Tenenbaum and Harry Pollard (1963), Ordinary Differential Equations, Harper and Row (reprinted in 1985 by Dover Publications). This is an inexpensive and encyclopedic treatment of the subject.

Vladimir I. Arnold (1973), Ordinary Differential Equations, MIT Press (as of 1991 in its 8th printing, translated from the Russian by Richard A. Silverman).  This is my favorite book of differential equations at a fairly elementary level.  What makes this book so remarkable is the use of geometric methods.  My columns will follow this book after a firm foundation has been laid with traditional methods.

Created by Mathematica  (March 11, 2004)