Differential Equation Model

Both exponential growth and exponential decay can be model with differential equations. Let's take a look how.

Recall that an exponential function is of the form y=ce to the kx. If you take the derivative with respect to x you get ce to the kx times k just from the chain rule. And of course this is just y. So dy dx equals k times y and that means that our original exponential function satisfies the differential equation dx, dy dx equals k times y, this is a very important differential equation. And so we say the general solution of this important differential equation dy dx equals ky is y=ce to the kx, the exponential functions. Same value of k, c would be some other constant, any constant would do.

Now, just to review. When k is greater than 0, we get exponential growth and when k is less than 0 we get exponential decay. And that goes for both of these equations. Here's an example, dy dx equals 0.1y. Here k is positive, so we get exponential growth. According to this formula the general solution is going to be y=ce to the k and k is 0.1x. So the whole family of functions y=ce to the 0.1x will be solutions of this differential equation and those are exponential growth functions. And depending on the c value you could get you know steeper 1 or a lower 1, c could be negative and so you could get 1 down here. But these are all different functions for different values of c.

Sometimes this k value is called the continuous growth rate and in that case it would be given as a percent. So the continuous growth rate here would be 10 percent. Now the important thing to know is that these exponential functions are solutions to this very important differential equation, dy dx=ky and we'll see applications of this in upcoming examples.