Separable Equations

Definition

Consider a differential equation of the form

$\frac{dy}{dt}=\frac{g(t)}{f(y)}$

Such an equation is called separable because we can separate the variables.

If $y(t)$ satisfies the equation, then

$$y^{\prime }(t)=\frac{dy}{dt}=\frac{g(t)}{f(y(t))}$$

Let $F(y)$ be an antiderivative of $f(y)$. Using the chain rule,

$$\begin{array}{r}\frac{d}{dt}F(y(t))=F^{\prime }(y(t))\ast y^{\prime }(t)\\ =f(y(t))\frac{g(t)}{f(y(t))}=g(t)\end{array}$$

So, $F(y(t))$ is an antiderivative of $g(t)$.

Formula

We get the following formula:

$$\int f(y)dy=\int g(t)dt+C$$