Posted by Soroban on August 09, 2002 at 15:10:51:
In Reply to: hmm... our textbook doesn't discuss this. posted by colin on August 09, 2002 at 10:42:06:
: : : : Read it and thou shall learn....
: : : So is it that you must be able to isolate variables of each type on opposite sides of the equals sign?
: : *****************************************
: : Suppose your ODE is in the form below:
: : f_1(x,y) dx = f_2 (x,y) dy
: : Now there is a relationship between"partial differential" of f_1 and f_2 that must be satisfied.
: ***************************************
: I've read and read in our DE textbook, and I can't find any mention of the use of partial derivatives to verify separable equations... How does that work?
I agree, Colin. I can't find any references
on a Partial Derivative test for Separability.
And I have looked in textbooks from my
undergraduate years up to the present.
For over 40 years, I've felt that there has been
no elegant "test" for Separability.
The following exchange has always struck me as
being rather immature and "whiny".
~ "Are the variables separable?"
~ "No."
~ "How do you know?"
~ "Because *I* can't separate them!"
Yet we're all forced to say it, aren't we?
One of my texts said: If the equation can be
written in form: M(x)dy + N(y)dx = 0, then it
is Separable. That's correct - but what if that
form is unattainble? How do we PROVE it?
To me, this is unnecessarily verbose. From
their form, we can obtain: dy/N(y) = -dx/M(x)
and we have LITERALLY "separated the variables".
Why do textbooks and professors shy away from
the actual physical Separation? Why must they
cloud the issue by writing equivalent equations?
Here's another equivalent equation. One text
said, if the DE is of the form: f(x)*g(y)dy +
h(x)*k(y)dx = 0, it is Separable. [Duh!]
I too would like someone to explain that
Partials Test. I tried to create one, but I can
only come up with necessary (but not sufficient)
conditions for Separability.