4.1.1. General concepts of first order differential equations

A first-order differential equation has the form F ( x, y, y ' ) or y' = f ( x, y ) , where y is an unknown function of the variable x .

The function y = φ ( x ) is called the solution of a differential equation if, by substituting y = φ ( x ) and its derivative into this equation, an identity is obtained.

The set of all solutions of a differential equation is called a general solution. It is represented as some function y = φ ( x, c ) ( c is a constant). With proper selection of the constant c, the function φ ( x, c ) defines any particular solution.

The task of finding a solution to a differential equation that satisfies the initial condition y ( x ₀ ) = y ₀ is called the Cauchy problem.