AP Calculus AB and BC glossary
General solution
The general solution of a differential equation is the entire family of functions that satisfy it, expressed with an arbitrary constant. Every specific solution is obtained by choosing a value for that constant.
Because the constant is unspecified, the general solution describes infinitely many curves at once, which is precisely what a slope field draws. Any one of them is a particular solution.
A first-order equation produces one arbitrary constant, so its general solution is a one-parameter family of curves. That count is why exactly one initial condition is needed to single out a solution.
Appears in: Unit 7: Differential Equations