Solve DE by method of substitution

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/