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Exponential growth

The equation $y' = ky$ says that the rate of growth is proportional to the quantity itself. The solution is an exponential function. The initial value $y(0)$ becomes the constant in front of the exponential function. If $k$ is positive, the solution grows, and if $k$ is negative, it decreases. You insert the time at the end. The model is used for unlimited growth and for radioactive decay.

Try it yourself

Logistic growth

y' = ky \Rightarrow y = Ce^{kx}

the general solution

C = y(0)

the initial value

Symbols

$k$	the growth constant
$C$	the initial value
$y$	the quantity

Example

$y' = 0.5y$ and $y(0) = 4$ gives $y = 4e^{0.5x}$ . Then $y(0) = 4$ and $y(2) = 4e^{1} \approx 10.87$ .

The constant

C

is the value when the exponent is zero, that is at

x = 0

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