Re: Quadratic to Linear solution

On Apr 30, 1:25 am, Paul <pkshree...@xxxxxxxxx> wrote:
Everyone,

Thank you very much! Please accept my apologies for such silly
mistakes of writing wrong solutions to usual quadratic and even linear
equations! I was looking at too many things at once while typing that
question, (some specialized equations, some general equations's
solutions, etc) so I ended up mixing them up in this post.

Thank you for correcting me! The book by Newcomb and reply by Dave
gave me very interesting alternative solution to quadratic equation
(one that I've never seen) that allowed me to do the limit function
toward linear limit. However I find it interesting that it still have
"sudden death" at the zero (two solutions suddenly to one solution).
While I will continue to work on this transistion issues, I want to
"promote" this discussion, the problem is the following,

partial differiental equations:

{dX}/{dT} = a X^2 + b X + c :: X and T is separable, and variable
"a" that goes from 1 to the limit of zero (as earlier question,
however now with dX/dT).
Now, we can solve this simply by integrating each side separably,

{dX}/{a X^2 + b X + c}= dT = Delta T

(oh hell, I'll just paste entire equation here in LaTeX typeset, I
don't know if it is possible to directly paste or attach to this
Usenet group?) I integrated it with X_{bottom} and X_{top} to create
"closed recursive" equation.
X_t=\frac{\sqrt{ac-\frac{b^2}{4}}X_b + (c+\frac{b}{2}X_b)\tan
((\sqrt{ac-\frac{b^2}{4}})\Delta T) }{\sqrt{ac-\frac{b^2}{4}}-(a X_b+
\frac{b}{2})\tan ((\sqrt{ac-\frac{b^2}{4}})\Delta T)}

Now, if I allow the variable "a" to go zero, of course this equation
goes nuts. The solution for {dX}/{b X + c}= dT = Delta T is
completely different using same approach as above (exponential
function).

I have tried using this quadratic "closed recursive" equation anyway
and allow the variable "a" to go exceedingly low but it created
absolutely no results. I suspect I have to develop a different method
of solving this, and possibly have to use something like Riemann's
solution or some type of "open" solution that doesn't completely fix
the type of curve for the solution?

Anyone can help me here? (Many thanks for previous remarks/comments/
help!)

Paul

Oh, I forget to mention...as this would be very important,..I use
complex numbers, and all variables are complex!
.

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