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6.8 Definition and general method for separable equations

A first-order differential equation is called separable if it can be written in the form

\frac{dy}{dx}=f(y)g(x)

where $f(y)$ , $g(x)$ are functions of one variable.

We rewrite the equation in the form $\frac{1}{f(y)}\frac{dy}{dx}=g(x)\;;$ then we integrate both sides with respect to $x$ to obtain

\int\frac{1}{f(y)}\frac{dy}{dx}\,dx=\int g(x)\,dx

and hence $\int\frac{1}{f(y)}\,dy=\int g(x)\,dx$ .

You may then have to rearrange to get a solution of the form $y=h(x)$ .