Exponential Models with Differential Equations

#7.8 Exponential Models with Differential Equations

Think of a video going viral and how quickly the number of views it gets increases over time. Or think about the last time who heard a rumor and how fast it spread around. These are real-world scenarios which can be explained through exponential models, a key concept in calculus that helps us understand things that grow or shrink really fast.

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An exponential graph which shows a relationship between popular video “Nyan Cat” and the number of views over time. A number which is steadily increasing.

Image Courtesy of Desmos

#🤔 What is a Differential Equation?

At the heart of exponential models are differential equations.

Imagine these equations as a mystery where you know how fast something is changing (like the speed of a rumor spreading), and you're trying to figure out the entire story (how many people will hear it over time). In calculus, these equations help us map out scenarios of rapid growth or decline.

The following equation is your go-to formula here.

$$dy/dt = ky$$

Rest assured, it's less complex than it looks.

• $dy/dt$ represents the rate of change of something over time, like the number of people watching a video or hearing a rumor.
• The $k$ in the equation is a constant that gives us the rate and nature of this change.
• A positive $k$ means things are growing or spreading (like more people watching the video).
• A negative $k$ means things are declining or fading away.

#✍🏾 Solving the Differential Equation

When the differential equation is solved, it gives us an important equation. Solve the steps below to find out what it is!

1. Start with the Differential Equation: We begin with the equation $dy/dt = ky$, as mentioned above.
2. Separate the Variables: To solve this e...