Differential Equation (DE)

Definition 1: Independent and Dependent Variables and Parameters

The dependent variable(s) of a DE are the unknown functions that we want to solve for i.e. f(x), y(x, t), etc. The independent variable(s) of a DE are the variable(s) that the independent variable(s) depend on i.e. x, t, etc. A parameter is a term that is an unknown but is not an independent or dependent variable i.e. a, b, α, β etc.

Definition 2: Order of a DE

The order of a DE is the order of the highest derivative.

Definition 3: ODEs and PDEs

A DE is an ordinary differential equation (ODE) if it only contains ordinary derivatives (i.e. no partial derivatives).

A DE is a partial differential equation (PDE) if it contains at least one partial derivative of a independent variable.

Definition 4: Linear and nonlinear DEs

A DE that contains no products of terms involving the dependent variable(s) is called linear.

If a DE is not linear then it is nonlinear.