Gavin's Calc II Class Clarification: Nov 8

Good question. In general, we just changed one constant to another. As an example, consider the equation y / 5 = k x (where k is a constant). If we're solving for y, we'd like to get rid of the one-fifth on the left-hand side, and so might multiply by it to get y = 5 k x. But 5k is just another constant, so we could say that this is the same as y = C x, for some new constant C (which happens to be equal to 5k). This is really all we were doing when playing with constants today -- any product or sum of constants is still a constant, so we can rename them with impunity. Let's look at that problem in class again. We had something like ln(|y+1|) = C + x²/2, for some constant C. Exponentiating both sides, |y+1| = e^C e^{x2 / 2}, but e^C is just another, positive constant. Let's call it k. Then |y+1| = k e^{x2 / 2}. To drop the absolute values on the left-hand side, we have to let the right-hand side be either positive or negative: y+1 = +/- k e^{x2 / 2}. But +/- k is just another constant, which can be positive or negative. Let's call it, ummm, G (for lack of a better letter), so that y = G e^{x2 / 2} - 1. Now, I like using C for my constants, so I'm going to rename G and call it C, getting y = C e^{x2 / 2} - 1. And, voila!, we're back to our solution. All we did in class was go through this process with a bunch of hatted-Cs.