On Wed, 06 Feb 2013 23:14:04 +0100, tuinkabouter
wrote:
On 2/6/2013 4:51 PM, Tim Wescott wrote:
On Wed, 06 Feb 2013 13:29:16 +0000, Bob Masta wrote:
On Wed, 06 Feb 2013 07:56:07 +0000, Martin Brown
wrote:
On 05/02/2013 19:26, Tim Wescott wrote:
On Tue, 05 Feb 2013 12:18:20 -0700, Jim Thompson wrote:
On Tue, 05 Feb 2013 13:16:39 -0600, Tim Wescott
wrote:
And, finally, if you do this a lot with linear, shift-invariant
difference equations, it pays to learn how to use the z-transform.
It simplifies things almost as much as the Laplace transform does
for linear time-invariant differential equations, and makes all of
this folderol much easier to remember.
Yep, It's all coming back to me... guess a solution and prove it fits
:-(
The guy who taught my second term of diff eqs clearly wanted us to
remember this for all time, because he made this a mantra. Nearly
every class meeting he would put some new form of differential
equation up on the board, and he'd say "Now, how do we solve this
differential equation?" then (because we didn't all shout it out in
unison) he'd answer himself: "We guess, and prove that we're right!"
That was 30 years ago. It's stuck with me, so I guess he met his goal
in my case.
I still recall my first serious brush with exotic differential equations
in the freshman year although for different reasons.
The lecturer was an internationally famous astronomer and brilliant
analytical solver of novel differential equations. The snag was that he
could not teach for toffee and merely demonstrated pulling rabbit out of
hat again and again and again. His coursework was impossible.
This particular course was so incomprehensible that after a while the
best of us went to the other stream of maths on group theory since it
was so much easier and the exam questions were likely to be possible to
solve in finite time. Some from the other course which then became badly
overcrowded then went to the ODE course so they could sit down.
Indirectly he probably contributed to the increased use of computers to
solve differential equations as we later moved into research.
I recall struggling through Diffy-Q, which was widely regarded as a "bag
of tricks" subject... you just had to figure out which trick to pull out
of the bag for each special case.
Then next term came Laplace Transforms, where we learned that everything
that needed doing could be done with simple algebra via Laplace... all
that Diffy-Q torture had been just for background information and
building character!
Using the Laplace transform is just a really versatile trick for
approximately solving real-world problems. Here's the reasoning that you
should keep in mind whenever you use it (or the z transform):
1: All real-world systems are nonlinear and time varying.
2: The Laplace transform only works on systems that are linear and time-
invariant.
3: Thus, I cannot use the Laplace transform to solve this problem.
4: But I can come _close_ by linearizing this here system
5: And now I can use Laplace!
This works great a whole lot of the time -- but it doesn't always, and
engineers who are steeped in Laplace (or the z transform) tend to forget
that they've skipped over steps 1-3, and did 4 without questioning why,
or when it is not valid to do so.
These discussion brings back some memories.
I hope i don't have bad dreams tonight.
Following to Laplace we where tortured with Hilbert transformations.
After university i never had to solve a differential- or integral equations.
Pure Laplace is a PITA, but the Heaviside short-hand version is a
beautiful engineering tool.
...Jim Thompson
--
| James E.Thompson, CTO | mens |
| Analog Innovations, Inc. | et |
| Analog/Mixed-Signal ASIC's and Discrete Systems | manus |
| Phoenix, Arizona 85048 Skype: Contacts Only | |
| Voice
480)460-2350 Fax: Available upon request | Brass Rat |
| E-mail Icon at
http://www.analog-innovations.com | 1962 |
I love to cook with wine. Sometimes I even put it in the food.