General and Particular Solutions of a Differential Equation

In earlier classes, solving an equation such as $x^2 + x - 2 = 0$ meant finding the real numbers that make both sides equal. Solving a differential equation is a different kind of task: instead of numbers, the solution is a function of the independent variable that satisfies the equation — along with its derivatives — when substituted in.

A differential equation can be satisfied by more than one such function. A solution written with an arbitrary constant (or several) still present in it is called a general solution — it represents an entire family of functions, one for every value the constant can take. The graph of any one member of this family is called a solution curve, or integral curve, of the differential equation. …