Ok, let's start by considering a problem like the one that we did in
class. We'll suppose that we're mixing industrial-sized quantities of
cookie dough and are interested in controlling the quantity of
chocolate chips in the mixing vat (of volume *V*) at all times.
The ideal quantity will be a function of time that we shall call
*a*(*t*). We expect the chips to be pumped into the vat at
a rate *r*(*t*), and dough is removed from the vat at a rate
*L*: then *a*(*t*) satisfies the differential
equation

*a*'(*t*) + *L a*(*t*) */ V* = *r*(*t*).
However, it may be that the machine that pumps chips into the vat
isn't quite accurate, and instead of pumping them in at a rate
*r*(*t*) pumps chips in at the rate *q*(*t*)
instead. Then the actual quantity of chips in the vat won't satisfy
the differential equation above---instead, it will be a quantity
*b*(*t*) that satisfies the differential equation

*b*'(*t*) + *L b*(*t*) */ V* = *q*(*t*).
Knowing that *a*(*t*) is not equal to *b*(*t*), we
might ask how far off it is---in particular, if we know something
about the magnitude of the difference *q*(*t*) -
*r*(*t*), we might ask if the difference between *b*
and *a* is small, or whether it might get arbitrarily large.
(*b*(*t*) - *a*(*t*))' + *L
(b*(*t*) - *a*(*t*)) */ V* =
(*q*(*t*) - *r*(*t*)), or

*z*'(*t*) + *L z*(*t*) */ V* =
(*q*(*t*) - *r*(*t*)).
Thus we have a differential equation for *z*(*t*) that we
can actually solve. How big can this deviation get? We saw in class
that (at least for a nice function) if the difference
(*q*(*t*) - *r*(*t*)) is bounded by some
number then we can be assured that the deviation *z*(*t*)
must similarly remain bounded---and that it can't change
discontinuously as the difference in forcing is changed. This is what
the book calls *continuity in the data*, and the book tells us
that this is a property of first-order differential equations
satisfying the conditions of the existence and uniqueness theorem.

In these equations, the *output* is *a*(*t*) (or
*b*(*t*))---they are the *response* of the system to
the *forcing*, or *input* given by *r*(*t*)
(or *q*(*t*)). The difference in this output is
*z*(*t*) = *b*(*t*) -
*a*(*t*). If we subtract the two equations above, we
get

Gavin's DiffEq Clarification 000218

Last Modified: Sun Feb 20 12:51:11 CST 2000

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