# Find the Error: Separation of Variables with Exponential Growth

It is a beautiful Spring Afternoon, you are sitting in the park, and you are happy. Your favorite Volleyball team is going to the Olympics, but that isn’t what is making you happy. Your sweetheart has just told you that you are getting a nice surprise for May Day, but that isn’t what is making you happy. A magazine company has just informed you that you are entered in a million-dollar sweepstakes, but that isn’t what is making you happy. You are happy because you have learned a new theorem:

If dy/dx = k y, then y = Aekx.

The nice thing about the theorem, you think, is that the converse is also true. In other words:

If y = Aekx then dy/dx = k y.

You hear some tinkly bells, and there is the ice-cream cart! You wave, and it comes towards you. But there is something wrong with the ice-cream vendor. He is wearing large dark glasses and an obviously fake beard and mustache. The bells are playing ghostly melodies in a minor key. You buy the ice-cream anyway (one of those ice cream bars on a stick), but after he has gone away, you notice this written on the wrapper: “Proof of the Theorem you learned today.”

You open the wrapper and to your amusement you see this written on the stick:

You are amused because there isn’t an error here. In fact, you might have seen this very proof in your calculus class. You eat your ice-cream, and then notice this written on the other side of the stick:

Why indeed? Find the error.