The notion of differential equation

Differential equations: Introduction to Differential equations

The notion of differential equation

Here is a description of a differential equation.

Differential equation
A differential equation is an equation whose solution is a function (or several functions). In addition to one or more unknown functions, one or more derivatives of these functions may occur in the differential equation.

A differential equation for one or more functions of a single (independent) variable is called an ordinary differential equation, abbreviated: ODE.
A differential equation for one or more functions of two or more variables, in which case the derivatives are partial derivatives, is called a partial differential equation, abbreviated: PDE.

The unknown in a differential equation is not a number but a function (of place, time, or both).

Computing an anti-derivative of the function #g# is the same as solving the differential equation \[y'=g\] Often we write #y'(t) =g(t)# for this equation. As is known from the chapter Integration, the general solution is denoted \[\int g(t)\,\dd t\]

This expression represents a differentiable function #y(t)# with derivative #g(t)# and is determined up to a constant factor. For example, if #g(t)=1#, the constant function #1#, then #y(t) = t# is a solution of the differential equation #y'(t)=1# and each solution is of the form #t+C#, where #C# is a constant.

The integral #\int g(t)\,\dd t# thus describes the general solution of the differential equation #y'(t) =g(t)#; it is a set of solutions described by means of a function rule in which one or more parameters (called integration constants), like #C# in the above example may occur.

Partial differential equations are not covered in this chapter: we almost only deal with differential equations with a single unknown function of one variable. Since the independent variable is often time (it is good practice to let the variable #t# stand for time), the ODEs are also known as dynamical systems.

The point of equations is to find solutions (or at least useful properties of the solutions). In general, those are found by rewriting the equation to simpler differential equations, or systems of equations (if possible even without derivatives). Just as in the case of equations without derivatives, it is convenient for this process to be able to indicate that two equations have the same solution.

Two ODEs having the same dependent unknowns and independent variables are called equivalent if they have the same solution.

The rate of growth in a population is determined by various biological processes and is directly influenced by the size of the population. Under ideal circumstances, as the size of the population increases, its rate of growth increases as well. A first-order differential equation can be used to express the rate of growth for a population as a proportion of its size:
\[y' - \dfrac{1}{3} y = 0 \]
Which question is being answered by solving this differential equation?

Solving this differential equation means answering the question:
Find a function \(y=f(x)\) so its growth \(\frac{\dd y}{\dd x}\), namely the first-order derivative \(f'(x)\), is equal to one third of the value #f(x)# of that function.

In our concrete example, the independent variable is #x#, time, and the unknown function \(y = f(x)\), also called the dependent variable, indicates how big the population is. The ODE expresses the mathematical intuition that the rate of population becomes higher as the population increases.

New example