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05 March 2004

Integral Curves and Differential Equations

by George E. Hrabovsky, President, MAST

What We Did Last Time

We explored the technique of integration by substitution.

Where We Will Go This Time

We will explore some of the ramifications of general differential equations.

Parameters in Differential Equations

Recall from, "Differential Equations in  General," that we found general solutions to the differential equations in the form

y(x) = ∫f(x) x + c .

and

| y(x) | = c ^(∫f(x) x ) .

Note the presence of the constant c in both solutions.  Let us say that we have a primitive function of x, y, and y/x, forming a homogeneous differential equation,

FormBox[RowBox[{F(x, y, y/x), =, 0.}], TraditionalForm]

Let us also assume that we can solve this for y/x,

y/x = f(x, y) .

If this function f(x, y) is continuous throughout some region R in the x - y plane, we can represent a solution of the form

(y/x) __0 = f(x_0, y_0),

for a point _0 within the region R. This solution determines a direction (the tangent of the solution at the point).  We can choose another point _1 within the region near _0 such that

(y/x) __1 = f(x_1, y_1) .

We can continue this process until we get something that looks like this.

[Graphics:HTMLFiles/index_19.gif]

We could link these points by line segments so that we get a broken curve.

[Graphics:HTMLFiles/index_20.gif]

If we bring the points closer and closer together we will eventually get a smooth curve.  If we think about this long enough we will see that if we start at a different initial point, we will get a different curve.  In this way the solution of a differential equation in general will produce a family of curves dependent upon the initial point.  Such a curve is called an integral curve since the process of solving a differential equation usually involves integration.  The initial point is, in part, determined by the value of the constant c discussed earlier.  Such a constant is called a parameter.

Though we are not yet ready to make this formal, there is a theorem attributed to the French mathematician Èmile Picard (1856-1941),

Picard's Theorem: If f(x, y) and ∂f/∂y are continuous functions on a closed region R, then through each point (x_0, y_0) in the interior of the region there will pass a unique integral curve of the differential equation y/x = f(x, y).

We will explore this connection between differential equations and geometry in more detail next time.

Mathematics Challenge from Last Time

The problem was to express partial derivatives of order higher than one.  Let me explain, a first order derivative is just a plain old everyday derivative,

y = sin x y/x = cos x .

A second order derivative is where you take the derivative of the derivative,

y = sin x y/x = cos x

                    ^2y/x^2 = -sin x .

A third order derivative is the derivative of the second order derivative, and so on.  For a function of two variables, say,

z = cos (x y),

we could have the following second order partial derivatives,

∂^2z/∂x^2, ∂^2z/(∂x ∂y), ∂^2z/∂y^2 .

Where,

∂^2z/∂x^2 = ∂/∂x (∂z/∂x)

            = ∂/∂x {∂/∂x[cos (x y)]}

            = ∂/∂x[-y sin (x y)]

            = -y^2 cos (x y),

and

∂^2z/∂y^2 = ∂/∂y (∂z/∂y)

            = ∂/∂y {∂/∂y[cos (x y)]}

            = ∂/∂y[-x sin (x y)]

            = -x^2 cos (x y),

and finally,

∂^2z/(∂x∂y) = ∂/∂x (∂z/∂y)

            = ∂/∂x {∂/∂y[cos (x y)]}

            = ∂/∂x[-x sin (x y)]

            = -x y cos (x y) - sin (x y) .

Mathematics Challenge

What 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 4, 2004)