# 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.