What are separable differential equations?

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What are separable differential equations?

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Answer 1 · 2014-09-07T15:25:28

A separable equation typically looks like:
$\frac{d y}{d x} = \frac{g (x)}{f (y)}$ ${dy}/{dx}={g(x)}/{f(y)}$ .

By multiplying by $d x$ $dx$ and by $f (y)$ $f(y)$ to separate $x$ $x$ 's and $y$ $y$ 's,
$\Rightarrow f (y) d y = g (x) d x$ $Rightarrow f(y)dy=g(x)dx$

By integrating both sides,
$\Rightarrow \int f (y) d y = \int g (x) d x$ $Rightarrow int f(y)dy=int g(x)dx$ ,
which gives us the solution expressed implicitly:

$\Rightarrow F (y) = G (x) + C$ $Rightarrow F(y)=G(x)+C$ ,
where $F$ $F$ and $G$ $G$ are antiderivatives of $f$ $f$ and $g$ $g$ , respectively.

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What are separable differential equations?

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