## Integrating Factors Found by Inspection

The following are the four exact differentials that occurs frequently.

1. $d(xy) = x \, dy + y \, dx$

2. $d\left( \dfrac{x}{y} \right) = \dfrac{y \, dx - x \, dy}{y^2}$

3. $d\left( \dfrac{y}{x} \right) = \dfrac{x \, dy - y \, dx}{x^2}$

4. $d\left( \arctan \dfrac{y}{x} \right) = \dfrac{x \, dy - y \, dx}{x^2 + y^2}$
5.

6. $d\left( \arctan \dfrac{x}{y} \right) = \dfrac{y \, dx - x \, dy}{x^2 + y^2}$

## Separation of Variables

Given the differential equation

$M(x, y)\,dx + N(x, y)\,dy = 0$   ←   Equation (1)

where M and N may be functions of both x and y. If the above equation can be transformed into the form

$f(x)\,dx + f(y)\,dy = 0$   ←   Equation (2)

where f(x) is a function of x alone and f(y) is a function of y alone, equation (1) is called variables separable.