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23 January 2004

A Fundamental Principle and Separation

by George E. Hrabovsky, President, MAST

What We Did Last Time

So far we have covered the basic definition of a differential equation and we have learned a few facts about the indefinite integral.

Where We Will Go This Time

I intend to write about the so-called Fundamental Theorem of Calculus and its ramifications.  Then I will go into the principle of Separation of Variables more carefully.

The Fundamental Theorem of Calculus

When talking about indefinite integrals the process can seem completely mystifying.  Such integrals are not easily calculated from first principles, like derivatives are. It takes a lot of creativity and knowledge to produce an indefinite integral without having a fairly complete table of derivatives in front of you.  Recall from last time that,

∫f(x) x = F(x) + C .

Where C is called the constant of integration and F(x) is called an antiderivative such that,

F ' (x) = F(x)/x = f(x) .

Another question is what this has to do with the formal and almost incomprehensible definition of a definite integral,

∫_0^af(t) t = Underscript[lim, Δ t0] Underoverscript[∑, i = 1, arg3] f(t_i) Δ t_i .

This is the point of The Fundamental Theorem of Calculus.  Here is the statement of the theorem,

∫_a^bf(t) t = ∫f(t) tUnderoverscript[], a, arg3]

                        = F(b) - F(a) .

Note that there is no constant of integration.  In plain language this states that the definite integral of a function is equal to the difference of the indefinite integrals at the limits of integration.

Separation of Variables

Last time I introduced a simple differential equation,

y/x = x^2 .

In order to solve an equation of this kind we want to integrate it.  So we need to get all of the same variables on their own side of the equal sign.  Here it is simple, we only need to multiply by x,

y = x^2x .

Now we integrate,

∫y = ∫x^2x .

Evaluating the left hand side we have,

∫y = F(y) + C .

The antiderivative can be found in any table of derivatives.  We ask the question, what function exists whose derivative is a 1?  The answer in this case is y, so

∫y = y + C .

How do we find the constant of integration?  We set the integral equal to some initial value y_0 and we set y equal to zero so that everything is constant, then the constant of integration is equal to the initial value,

∫y = y + y_0 .

We now turn to the right-hand side of the equation,

∫x^2x = F(x) + C .

What function has x^2 as its derivative? The answer is,

∫x^2x = x^3/3 + C .

We find the constant of integration as we did above.  We set the integral equal to an initial value of x, and then set x = 0, thus we get,

∫x^2x = x^3/3 + x_0 .

Putting these results together we get,

y + y_0 = x^3/3 + x_0

or,

y   = x^3/3 + x_0 - y_0 .

Mathematics Challenge from Last Time

This has been done in the discussion above.

Mathematics Challenge

Solve the differential equation given above by definite integration using the Fundamental Theorem of Calculus.

Created by Mathematica  (January 23, 2004)