# Solving Differential Equations by Separating Variables

Standard

Differential Equations are equations that involve a function and its derivatives. Sometimes they can be solved using a technique called separating variables. This only works for some differential equations.

Separable differential equations are equations of the form:
$N(y)y'=M(x)$ which can also be written as:

$N(y)\frac{dy}{dx}=M(x)$

#### Example:

$\frac{dy}{dx}=y^2\cos(x)$

1. First we have to convert this to a separable differential equation. To do this multiply both sides by $\frac{1}{y^2}$:
$\frac{1}{y^2} \frac{dy}{dx}=\cos(x)$

2. Multiply both sides by dx:
$\frac{1}{y^2}dy=\cos(x)dx$

3. Integrate both sides:
$-\frac{1}{y}+C_1=\sin(x)+C_2$
(the C1 and C2 are arbitrary constants of integration):

4. Combine the arbitrary constants:
$-\frac{1}{y}=\sin(x)+C$

5. Take the negative reciprocal of both sides:
$y = -\frac{1}{\sin(x)+C}$

As you can see differential equations are separable if you can separate the variables and then integrate to solve.