ODE Exponential growth

$y={y}_{0}{e}^{\lambda t}$

step=
t-min= t-max=
y-min= y-max=
λ-min= λ-max=
λ=
t0= y0=

Model

The model of exponential growth is based on that a stock y (eg. a population) is all the more increased the greater the inventory itself. That in the model with the magnitude of the population continues to increase the growth rate.

Limitations of the model

The model does not account limiting factors such as only a finite available resources. A model that takes this into account leads to the model of logistic growth.

Half-life or doubling time T

The half-life or doubling time T is the period in which the stock is doubled or halved.

Exponential growth:

$T = ln 2 λ$

Exponential decay:

$T = ln 12 λ$

Application Examples

• Growth of populations

• Absorption of light

• Compound interest

Differential equation of the exponential growth

The continuous exponential growth by a homogeneous linear differential equations with constant factor λ described.

$y′t= d y d t =λyt$

The equation states that the change of the stock is proportional to the component itself. λ is the proportionality constant.

$y′y=λ$

Division by y

$lny′=λ$

Applying the chain rule

$lny=λ∫dt=λt+C$

Integration

$y=y0eλt$

General solution of the equation for exponential growth with the constant y0 is the initial value. For λ > 0 exists an exponential growth process and and for λ < 0 an exponential decay process.