Explicit and implicit differential equations

Summary

An ordinary differential equation of order $n$ is explicit when it has the form $y^{(n)} = f\left(x, y, y', \dots, y^{(n-1)}\right)$ for a given function $f$. It is implicit when it has the general form $F\left(x, y, y', \dots, y^{(n)}\right) = 0$ and is not solved for $y^{(n)}$. Each explicit equation is also implicit, with $F = y^{(n)} - f$.

Remarks

The equation $y' = x^2 + y^2$ is explicit. The equation $(y')^2 + y^2 = 1$ is implicit. If $\partial F / \partial y^{(n)}$ is not zero at a point, the implicit function theorem gives an explicit equation near that point. The basic existence theorems, such as the Picard–Lindelöf theorem, are for explicit equations.

Explanations

L1 human

An explicit equation tells you the highest derivative directly, as a formula in the lower ones. An implicit equation only gives a relation, and you may have to solve it for $y'$ first. For $(y')^2 + y^2 = 1$ you get two explicit equations, $y' = \sqrt{1 - y^2}$ and $y' = -\sqrt{1 - y^2}$.