Exact Equations

Multivariable functions

A DE in the form of:

M (x, y) + N (x, y) \frac{d y}{d x} = 0

is called exact if there exists a function $φ (x, y)$ such that

(total derivative) \frac{d φ}{d x} = M (x, y) + N (x, y) \frac{d y}{d x}

Exact equations

We can find $φ (x, y)$ by integrating $M$ with respect to x, or $N$ with respect to y

$M$ and $N$ may not contain all the information about the variable not integrated wrt

Aligning $M$ and $N$ to find the missing terms we get the full form of $φ$
Total Derivative

Using Integrating Factors to Transform "Almost Exact" DEs into Exact

Start with a non-exact DE:

M (x, y) + N (x, y) \frac{d y}{d x} = 0

Multiplying by an integrating factor $μ (x, y)$ :

μ (x, y) M (x, y) + μ (x, y) N (x, y) \frac{d y}{d x} = 0

For the DE to be exact, we need:

\frac{\partial}{\partial y} (μ (x, y) M (x, y)) = \frac{\partial}{\partial x} (μ (x, y) N (x, y))

If $μ$ is a function of $x$ only