In solving ordinary differential equation
………. (*)
(where are constants)
we use the method of substitution, let
Then we have and hence
Therefore, (*) can be further reduced as a second order linear differential equation with constant coefficients:
, where
The first question come out from students’ minds: how can you think about the substitution ? Sorry, I cannot answer. This should be a great idea from someone(s) in the past, but how could he/she think about that? Well, you may explore the historical facts and tell me later. Second question, as asked by a student today morning, Chan, is it always possible to substitute
? Good question.
is different from
in general, at least
must be positive but
may be negative.
OK, let’s consider a concrete example from a textbook.
Solve
………. (#)
In the solution, it wrote immediately and, after a series of mechanical procedure, it yields
Obviously, the above is valid only when .
But, there should not be any restriction on in the equation (#).
So, what is the solution to (#) indeed? Is it simply add absolute signs to the above and yield
?
Urm…students, you may try on your own:
when , let
. See what you will obtain?
OK? Do you obtain something like
for
?
(Please help me to debug, because I just did it in a hurry…)
For better setting of the question, we may post the question as
Solve
for
.
………………………….
[OT]
To 7B students, please refer to the following post for the question I’d mentioned in the lesson today:
https://johnmayhk.wordpress.com/2007/10/05/alpm-past-paper-1998-paper-ii-q11/
The substituition using exponential function is very useful is solving linear ODE.
Another issue which use a similar method is the Cauchy-Euler equation.
For further reading,
http://en.wikipedia.org/wiki/Linear_differential_equation
http://en.wikipedia.org/wiki/Cauchy%E2%80%93Euler_equation
迴響 由 Justin — 2009/10/20 @ 1:34 上午 |
Thank you Justin for your links. As mentioned in the second webpage you’d given, the trial solution is
in solving the Euler–Cauchy equation, however, the solution turns up something like
, it may be quite puzzling.
迴響 由 johnmayhk — 2009/10/20 @ 12:47 下午 |
我記得學過
Liberal Studies, the Liberal Us!
http://wp.me/PyvmP-3b
迴響 由 lslu — 2009/10/20 @ 9:53 上午 |