Explicit and implicit differential equations

Builds on Ordinary differential equation

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.

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Differential equation concept
Function concept
Ordinary differential equation definition
Explicit and implicit differential equations concept

Tag	`08WWTN738R`
URI	`m2://08WWTN738R`
Kind	concept
Writing	W0 stub
MSC	34A09
Uses	Ordinary differential equation

Explicit and implicit differential equations

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Remarks

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