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13 February 2004

Differential Equations in General

by George E. Hrabovsky, President, MAST

What We Did Last Time

We continued our exploration of separation of variables in first-order differential equations.

Another Note of Apology

I would like to thank Rou Olund for pointing out that the expression

ln(a + b) = ln a ln b,

is backwards!  It should read

ln a + ln b = ln (a  b),

so the answer

y(x) = ln ((x^2 - 1)/2 + (x - 1^3)/3) .

is wrong and it should be,

y(x) = ln ((x^2 - 1)/2 + (x - 1^3)/3 + ) .

Thanks for pointing this out!  The only thing I dislike more than making a mistake, is not catching it in time.

Where We Will Go This Time

I will go from the specific, separable equations, to a discussion of the general linear first-order differential equation.

Linear Differential Equations in General

We have been talking about differential equations for a while now, but we have been discussing separable equations.  We have not talked about differential equations in general, at the linear equations.  The first thing to do is establish what is meant by the term linear differential equation.  A differential equation is linear if the highest order derivative is a linear function of the lower order derivatives.  This is a linear differential equation

y/x = a(x) y + b(x) .

This is not,

y/x = a(x) y/x + b(x) .

We will be concerned, for now, with the simple case

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

Any equation of the form  = 0 is called a homogeneous equation.
    We can solve this by writing,

y/x = f(x),

and then separating the variables to get

y = f(x) x .

We then integrate both sides

∫y = ∫f(x) x + c .

or,

y(x) = ∫f(x) x + c .

    Another common form of homogeneous differential equation is

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

This can be rewritten,

y/x = f(x) y .

We separate variables,

y/y = f(x) x

and integrate to get,

∫y/y = ∫f(x) x + c_1 .

We know (or can find in any table of integrals) that

∫y/y = ln | y(x) |,

so

ln | y(x) | = ∫f(x) x + c_1 .

To solve this, we take the exponential function of both sides,

| y(x) | = ^(∫f(x) x + c_1) .

We know from basic algebra that,

^(a + b) ^a^b

so,

| y(x) | = ^(∫f(x) x + c_1)

FormBox[RowBox[{              ... #63309;^(∫f(x) x), RowBox[{, ^, RowBox[{c, _, 1.}]}]}]}]}], TraditionalForm]

Since ^c_1 is a constant, we will rename this c, so

| y(x) | = c ^(∫f(x) x ) .

Thus we have solved the general homogeneous differential equation of first order.

Mathematics Challenge from Last Time

The entire article was solution of the challenge from last week.

Mathematics Challenge

Speculate on how to take the derivative of a function of more than one variable.

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 differnetial 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 enixpensive and encyclopaedic 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 failry 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  (February 12, 2004)