Differential equations have two kinds of solutions: general and particular.
The general solution to a differential equation is the collection of all solutions to that differential equation. A general solution will usually contain some undetermined constants.
Sample Problem
y = 4x + C is the general solution to the d.e.
y ' = 4.
Sample Problem
y = 3x2 + Bx + C is the general solution to the d.e.
y " = 6.
A particular solution to a differential equation is a solution with all the constants filled in.
Sample Problem
The function y = 4x + 2 is a specific solution to the d.e.
y ' = 4.
Initial value problems usually have a particular solution only, because the initial condition(s) force us to pick values for the constant(s) in the general solution.