Understanding Basic Differential Equations

Summer 2021 ~ @TitouanCh
Differential equations are equations where the variable is a function. They are a powerful tool to model the world around us...

Differential equations of the first order

The order of a differential equation corresponds to the highest degree of derivation found in the equation. For example : y = y' + 1, the order is equal to 1 because y' is the highest degree of derivation found in the equation.
In this article, we'll only look at differential equations of the first order as they are the most simple.

First order equation example : y'(t) = αy(t)

We graphed the equation : y'(t) = αy(t) above. With α a parameter.
Sur le graphique, nous observons des courbes rappelant la fonction exponentielle.

To graph the equation, we first chose random points on the graph with the y-axis representing y and the x-axis representing t. Then we draw vectors (y, y') to represent were the equation will tend to evolve to next.
The drawn vectors help us gain a better understanding of the equation and possibly help us find a solution.
We can observe curves reminding us of exponential functions. Indeed the solution to the equation is y(t) = exp(α + t).

What can differential equations be used for ?

Despite its simplicity, the equation we've just studied has many application. For example, we can use it to model population dynamics.
We can use it to model the growth of a population.
We'll have :
y' : correponding to the speed at which the population is growing.
y : corresponding to the population size.
α : corresponding to the population's growth factor.
With our equation : y'(t) = αy(t), our growth speed (y') will be proportional to our population size (y) and our growth factor (α).

Differential equations are incredible tools to study phenomenons where the rate of change is affected by its parameters.