First-Order Differential Equations

A first-order differential equation is a type of differential equation that contains a function and its first derivative. In other words, it involves only the first derivative of the function, not higher-order derivatives.

General Form

The general form of a first-order differential equation is:

dy/dx = f(x, y)

where:

  • dy/dx is the first derivative of y with respect to x,
  • f(x, y) is a function of x and y.

Solving First-Order Differential Equations

Solving a first-order differential equation involves finding a function that satisfies the equation. The method to solve it depends on the form of the differential equation. Here are a few common types and their methods of solution:

  1. Separable Equations: These are first-order differential equations that can be written in the form dy/dx = g(x)h(y). They are solved by "separating" the variables x and y on opposite sides of the equation and then integrating both sides.

  2. Linear Equations: These are first-order differential equations that can be written in the form dy/dx + p(x)y = q(x). They are solved using an integrating factor.

  3. Exact Equations: These are first-order differential equations that can be written in the form M(x, y)dx + N(x, y)dy = 0, where M and N satisfy certain conditions. They are solved by finding a function whose total differential equals the left-hand side of the equation.

Applications

First-order differential equations are used in a wide range of applications, including physics, chemistry, engineering, and biology. For example, they are used to model population growth in biology, the behavior of circuits in electrical engineering, and the motion of objects in physics.