Chapter 5, Problem 1CCE

What differential equation is key to solving exponential growth and decay problems? State a result about the solution to this differential equation.

To determine

The differential equation which is the key to solve the exponential growth and decay problems, and state result about the solution to this differential equation.

Answer to Problem 1CCE

Solution:

The differential equation to solve exponential growth or decay function $y=P(t)$ is $P\text{'}(t)=kP(t)$, i.e., $y\text{'}=ky$ and its solution is $y=C{e}^{kt}$ for some constant $C$, where $k$ is a constant of proportionality.

Explanation of Solution

Given information:

The instruction to state the differential equation which is the key to solve the exponential growth and decay problems and its solution.

Explanation:

Let $P(t)$ denote the population at time $t$, then the differential equation which is the key to solve the exponential growth and decay problems is given by:

$P\text{'}(t)=kP(t)$, where $k$ is a constant of proportionality.

Let $y=P(t)$, then the equation becomes $y\text{'}=ky$.

The function $y=C{e}^{kt}$ for some constant $C$, satisfy the differential equation $y\text{'}=ky$.

Therefore, $y=C{e}^{kt}$ is solution of $y\text{'}=ky$, where $k$ is a constant of proportionality.