Differential Equations for Exponential Growth and Decay

The exponential growth and decay formulas can be seen two different ways: as formulas, which they are called on a day-to-day basis; as differential equations, the other way to look at the formulas, which I will explain in just a second.

The formula for exponential growth/decay is shown below, where k is positive if the formula is for exponential growth, and negative for the opposite.

$f\left(t\right)=ce^{kt}$ , where c= value of f at (t=0), and k is the rate of growth.

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-Differential Equation: when you use a function and its derivative(s) to find an unknown function, like using

$y\left(x\right)=2x$

$y'\left(x\right)=f\left(x\right)$

to see what f(x) equals.

We revisit exponential growth and decay from the perspective of differential equations: namely, the exponential differential equation, $y' = ky$, where $k$ is a constant (sometimes called the relative growth constant). The general solution is $y(t) = ce^{kt}$, which exhibits exponential growth when $k$ is positive, exponential decay when $k$ is negative, and neither when $k=0$. Here $c = y(0)$ is determined by the initial conditions (or sometimes other data).