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23 January 2004

Initial-Value Problems

by George E. Hrabovsky, President, MAST

What We Did Last Time

We explored the method of separation of variables more deeply by applying the fundamental theorem of calculus.

Where We Will Go This Time

I intend to develop an initial-value problem for a separable differential equation.

Separation of Variables, Part Two-- An Introduction to Dummy Variables

Last time we solved a differential equation by the method of separation of variables.  Such an equation is called separable. We found the general solution of the separable equation. Recalling that the symbol ⟹ means that whatever is on the left-hand side implies whatever is on the right-hand side,  we can then write,

^yy/x - (x + x^2) = 0 ⟹y (x) = ln[ (x^2/2 + x^3/3) -    (x_0^2/2 + x_0^3/3)] .

Let's say that we want to see what happens when the initial value of y has a specific value y_x_0.  For our purposes, let's assign x_0 = 1, y_1 = 1.  We begin by again separating the variables for the equation to get,

^yy = (x + x^2) x .

We integrate the right hand side as we did before to get

∫_1^x (x ' + x '^2) x ' = (x^2 - 1)/2 + (x - 1^3)/3

Now we integrate the left hand side, too,

∫_1^y^y ' y ' = ^y -  .

Thus,

^y -  = (x^2 - 1)/2 + (x - 1^3)/3 .

Or,

^y = (x^2 - 1)/2 + (x - 1^3)/3 +  .

We solve this by taking the natural logarithm of both sides,

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

Since we know the rule that

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

and we know

ln() = 1

then, the solution to the initial value problem is

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

Mathematics Challenge from Last Time

The entire article was about how to do an initial value problem.  It is an instructive exercise to do several initial-value problems for the same general solution.

Mathematics Challenge

Speculate on how to solve a differential equation in general, say the equation,

y/x = f(x) .

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 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 5, 2004)