# Exact Equations | Equations of Order One

The differential equation

$M(x, y) \, dx + N(x, y) \, dy = 0$

is an exact equation if

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

**Steps in Solving an Exact Equation**

- Let $\, \dfrac{\partial F}{\partial x} = M \,$.
- Write the equation in Step 1 into the form

$\displaystyle \int \partial F = \int M \, \partial x$and integrate it partially in terms of x holding y as constant.

- Differentiate partially in terms of y the result in Step 2 holding x as constant.
- Equate the result in Step 3 to N and collect similar terms.
- Integrate the result in Step 4 with respect to y, holding x as constant.
- Substitute the result in Step 5 to the result in Step 2 and equate the result to a constant c.