when the two roots s1 and s2 that are different of the homogenius equation
in normal diff theory i do

[math]
v_c (t)=Ae^{s_1t}+Be^{s_2t}
[/math]

why why in basic circuit theory we do
[math]
v_c (t)=Ae^{-|s_1|t}+Be^{-|s_2|t}
[/math]
(this is a standard latex code in this forum its not working
so you can copy it into here
http://www.codecogs.com/components/e...tioneditor.php
and you will get my clear equations

You can always have negative signs or positive signs if you wanted to. It works either way and doesn't change anything.

It's just that we know from observing circuits that the voltage is decaying (which is why the exponent is negative in the solution). So we put the negative sign in there beforehand so when we solve for s1 and s2 we get a nice positive number, instead of a negative number. If we didn't put the negative sign there we would just get a negative number for s instead.