The Method of Separation of Variables for First Order Differential Equations.

Use the method of separation of variables to help simplify differential equations of the form: $$\frac{dy}{dx}=f\left(x\right)g\left(y\right)$$ Assuming $g\left(y\right)\ne 0$, then $$\frac{1}{g\left(y\right)}\frac{dy}{dx}=f\left(x\right)$$ Integrate with respect to $x$ $$\int \frac{1}{g\left(y\right)}dy=\int f\left(x\right)dx+C$$ where $C$ is a constant.

Solve each integral, perhaps using the integration flow chart.

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If $g\left(y\right)=0$, then $$\frac{dy}{dx}=0$$ and $$y=\int 0dx+C$$ $$y=C$$