I am currently taking a calculus 3 class in college, and my teacher did an intriguing problem about a tsunami in class which took up about 4 full white boards. Anyways, at a certain point in the problem, we arrived at $ |{u-2 \over u+2 }| = e^{2Cx}$
Now, of course we can say that this is the same as $ |{u-2 \over u+2 }| = e^{2x}e^C$, but this is where things get a little funky. Since $e^c$ is just another constant, we can rewrite the equation as $ |{u-2 \over u+2 }| = e^{2x}C$
Alright, so eventually we got the answer $ C= -1$. This did indeed solve our problem, but then I noticed the serious issue that occurs when we go back and apply this to our equation. At the point $ |{u-2 \over u+2 }| = e^{2Cx}$, $C = -1$ is perfectly acceptable, since $e^x$ can never be negative. But our issue comes about when we redefine $e^C$ as a new constant $C$. With our answer $C = -1$, we wind up getting $ |{u-2 \over u+2 }| = -e^{2x}$
Uh-oh! Clearly this will not do, since it's impossible! Evidently, the error occurred when we redefined the constant -- but why? My teacher's solution was to simply remove $|...|$ from the equation, but this seems to be an illogical and unsatisfactory answer. So then, what went wrong? Can redefining a constant lead to fatal errors in math? Thanks for your answers.
Note that we can express $\exp(2x+C)=\exp(C)\exp(2x)=D\exp(2x)$
but we can't expressed $\exp(2Cx)$ as $D\exp(2x)$. There is indeed an error in your simplification.
You might also like to use different notation as you perform a substitution to avoid confusion.