Differential Equations of the form $\frac{dy}{dx} = f(x)g(y)$

We have learned how to solve differential equations which depend on either $x$ or $y$, but not both at the same time. It turns out that we can combine the techniques we have seen already to solve what are called separable differential equations. As the name suggests, a separable differential equation is one that can be separated into parts dealing with only one variable in each part. For example, we could use this technique to solve the differential equation

$$\frac{dy}{dx} = \frac{x}{y}$$

In this case, we now have a derivative that depends on the solution and the value of $x$ at the same time.

We will again solve this using the Change of Variables Theorem, so the solution will look a lot like what we did for the example $\frac{dy}{dx} = y$.

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