ABSTRACT
Existence Theorems 12.1.1-1. Equations solved for the derivative. General solution.
A first-order ordinary differential equation* solved for the derivative has the form
y′x = f (x, y). (12.1.1.1)
Sometimes it is represented in terms of differentials as dy = f (x, y) dx. A solution of a differential equation is a function y(x) that, when substituted into
the equation, turns it into an identity. The general solution of a differential equation is the set of all its solutions. In some cases, the general solution can be represented as a function y = ϕ(x, C) that depends on one arbitrary constant C; specific values of C define specific solutions of the equation (particular solutions). In practice, the general solution more frequently appears in implicit form, Φ(x, y, C) = 0, or parametric form, x = x(t, C), y = y(t, C).
