# Renaming constants

1. Feb 2, 2016

### Joseph1739

Suppose I'm at this point in solving a differential equation and the initial condition is Q(0) = Q0
-ln|25-Q| + c1 = rt/100 + c2
Then if I combine c2-c1, I can rename it to c, we have:
-ln|25-Q| = rt/100 + c
Now if I multiply the equation by (-1), I get:
ln|25-Q| = -rt/100 - c
If I let -c = C:
ln|25-Q| -rt/100 +C

But if I rewrite -c = C, all the signs are reversed when I solve for Q. Also solving for the constant, my book kept the -c, and got c = Q0-25, but when I rewrite -c to C, I get C = 25-Q0.

So my question is, when can I rename constants? When I combined two constants it is okay to to rename the constant, but why is it incorrect when I negate a constant and rename it?

2. Feb 2, 2016

### Staff: Mentor

You can simplify things a bit by including the constant only on one side (the right side).

You lost an = in the line above.
Whenever you want to.
If -c = C, and the book shows c = $Q_0 - 25$, then C = $-(Q_0 - 25) = 25 - Q_0$.

3. Feb 2, 2016

### the_wolfman

Looking at your last 2 equations, at $t=0$ you either get $ln\left|25-Q_0 \right| = -c$ or $ln\left|25-Q_0 \right| = C$. These two results agree with your definition $c=-C$.