An algebraic equation asks for numbers that satisfy a relationship. A differential equation asks for a function whose derivatives satisfy a relationship.

An ordinary differential equation (ODE) involves an unknown function of one independent variable and one or more of its derivatives.

For example,

$$\frac{dy}{dt}=2y$$

asks for functions whose rate of change is always twice their current value.

Solutions are functions

The function

$$y(t)=Ce^{2t}$$

satisfies the equation because

$$\frac{dy}{dt}=2Ce^{2t}=2y.$$

The arbitrary constant $C$ means the differential equation describes a family of solutions.

Order

The order of an ODE is the highest derivative that appears. Thus

$$y''+4y=0$$

is a second-order equation.

Initial conditions

Additional information can select one solution from the family. If

$$y(0)=3,$$

then the condition determines the value of an otherwise free constant.

A second-order equation generally requires two independent initial conditions, such as initial position and velocity.

Differential equations provide a natural language for physical laws because many laws specify how a system's current state determines its rate of change.