Category:Exact Differential Equations

This category contains results about Exact Differential Equations.
Definitions specific to this category can be found in Definitions/Exact Differential Equations.

Definition 1

An exact differential equation is a first order ordinary differential equation in which the total differential is equal to zero:

$\dfrac {\partial f} {\partial x} \rd x + \dfrac {\partial f} {\partial y} \rd y = 0$

Definition 2

Let a first order ordinary differential equation be expressible in this form:

$\map M {x, y} + \map N {x, y} \dfrac {\d y} {\d x} = 0$

such that $M$ and $N$ are not homogeneous functions of the same degree.

However, suppose there happens to exist a function $\map f {x, y}$ such that:

$\dfrac {\partial f} {\partial x} = M, \dfrac {\partial f} {\partial y} = N$

such that the second partial derivatives of $f$ exist and are continuous.

Then the expression $M \rd x + N \rd y$ is called an exact differential, and the differential equation is called an exact differential equation.