Solving Diff Eqns: Renaming Constants & Reversing Signs

[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?

 

[Mark44]

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

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

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

You lost an = in the line above.

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?

Whenever you want to.

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?

If -c = C, and the book shows c = $Q_0 - 25$, then C = $-(Q_0 - 25) = 25 - Q_0$.

 

[the_wolfman]

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?

Check your math. Your calculation of c and -C are wrong.

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.

You can always define a new constant as a combination of multiple constants.