The Differential Equation Model for Exponential Growth - Concept
If a function is growing or shrinking exponentially, it can be modeled using a differential equation. The equation itself is dy/dx=ky, which leads to the solution of y=ce^(kx). In the differential equation model, k is a constant that determines if the function is growing or shrinking. If k is greater than 1, the function is growing. If it is less than 1, the function is shrinking.
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.