The model of exponential growth is based on the fact that the larger the stock y (e.g. a population), the more strongly it multiplies the larger the stock itself. This means in the model that the growth rate increases with the size of the population.

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

Differential equation of exponential growth:

$$y\prime \left(t\right)=\lambda y\left(t\right)$$

$$\lambda \phantom{\rule{0.3em}{0ex}}:\phantom{\rule{0.3em}{0ex}}\mathrm{Exponential\; growth\; rate}$$

With the growth function for the general inital values t_{0} and y_{0} = y(t_{0})

$y\left(t\right)={y}_{0}{e}^{\lambda \left(t-{t}_{0}\right)}$

The half-life or doubling time T describes the time in which the stock doubles or halves.

Exponential growth:

$$T=\frac{ln2}{\lambda}$$

Exponential decay:

$$T=\frac{ln\frac{1}{2}}{\lambda}$$

Application Examples:

Growth of populations

Radioactive decay

Absorption of light

Compound interest

The continuous exponential growth is described by a linear homogeneous differential equation with the constant factor λ

$$y\prime \left(t\right)=\frac{dy}{dt}=\lambda y\left(t\right)$$

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

$$\frac{y\prime}{y}=\lambda $$

Division by y

$$\left(\mathrm{ln}y\right)\prime =\lambda $$

Applying the chain rule

$$\mathrm{ln}y=\lambda \int dt=\lambda t+C$$

Integration

$$y={y}_{0}{e}^{\lambda \left(t-{t}_{0}\right)}$$

Dissolving and replacing the initial condition t_{0}, y_{0} yields the solution for exponential growth. For λ > 0 there is an exponential growth process and for λ < 0 an exponential decay process.

Print or save the image via right mouse click.

Here is a list of of further useful sites:

Index Logistic growth function Function Plot Normal Distribution Plot Trigonometry Curve fit calculator Damped Vibration Plot Derivative calculus Partial derivatives and gradient Tangent (tan) Plot Beat frequencies Plot