Explain how these 2 definitions are different - Differential Equation

Edwards & Penney, Calculus with Analytic Geometry, Early Transcendentals (p412) - A first-order differential equation is a differential equation that can be written in the form dy dx = g(x, y) where x denotes the independent variable and y = y(x) is the unknown function. A solution...is a function y = y(x) such that y 0 (x) = g(x, y(x)) for all x in some appropriate interval I.

Finney, Demana, Waits, and Kennedy, Calculus: Graphic, Numeric, Algebraic (p322) An initial condition determines a particular solution by requiring that a solution curve pass through a given point. If the curve is continuous, this pins down the solution on the entire domain. If the curve is discontinuous, the initial condition only pins down the continuous piece of the curve that passes through the given point. In this case, the domain of the solution must be specified.