Problem 22 - Essential Calculus

Question

Give the differential equation that models exponential growth and decay.

Step-by-Step Solution

Answer

For exponential growth, the differential equation is \[ \frac{{dy}}{{dt}} = ky \], and for exponential decay, it is \[ \frac{{dy}}{{dt}} = -ky \]

Step 1: Understand Exponential Growth Model

The differential equation that models exponential growth is \[ \frac{{dy}}{{dt}} = ky \], where \( y \) is the quantity we are interested in, \( t \) is the time, \( k \) is the proportionality constant, and \(\frac{{dy}}{{dt}}\) is the rate of change of the quantity.

Step 2: Understand Exponential Decay Model

The differential equation that models exponential decay is very similar to the growth model. The difference is in the sign of the proportionality constant. The model for exponential decay is \[ \frac{{dy}}{{dt}} = -ky \], meaning the rate of change is proportional to, but in the opposite direction of the quantity.