First-Order Differential Equation

A first-order differential equation is a differential equation that involves only the first derivative of an unknown function, and no higher-order derivatives. These equations are of the form:

dy/dx = f(x, y)

First-order differential equations are widely used in various fields such as physics, engineering, and economics to model a wide range of phenomena, from population growth to fluid dynamics.

There are several techniques for solving first-order differential equations, including separation of variables, integration factors, and undetermined coefficients. The choice of technique depends on the structure and complexity of the given equation.

Some first-order differential equations have unique solutions, while others may have multiple solutions or no solution at all. The existence and uniqueness theorem provides conditions under which a first-order differential equation has a unique solution.