Exponential growth and decay

Example

Let $k$ and $y_0$ be real numbers. The initial value problem $y'(t) = k\, y(t)$ , $y(0) = y_0$ has the unique solution $y(t) = y_0\, e^{k t}, \qquad t \in \mathbb{R}.$ If $k > 0$ and $y_0 > 0$ , the solution grows: this is exponential growth. If $k < 0$ and $y_0 > 0$ , the solution decreases to $0$ : this is exponential decay.

Remarks

For $k > 0$ , the doubling time is $\ln 2 / k$ : we have $y(t + \ln 2 / k) = 2\, y(t)$ for each $t$ . For $k < 0$ , the half-life is $\ln 2 / |k|$ . Radioactive decay follows this law. Uniqueness: if $y$ is a solution, then the derivative of $y(t) e^{-k t}$ is $0$ , so $y(t) e^{-kt}$ is the constant $y_0$ .

Reading path

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Differential equation concept
Function concept
Interval concept
Ordinary differential equation definition
Solution of an ordinary differential equation definition
Initial value problem definition
Exponential growth and decay example

Where this leads

10 pages build on this one.

Tag	`00SVP6W3CB`
URI	`m2://00SVP6W3CB`
Kind	example
Writing	W2 usable
MSC	34A05
Instance of	Separable equation, Linear first-order equation
Uses	Initial value problem
Same as	Wikidata Q582659 Wikipedia Exponential growth

Exponential growth and decay

Example

Remarks

Reading path

Where this leads

Further reading

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