# How do you solve separable differential equations with initial conditions?

Oct 1, 2014

A separable equation typically looks like:

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{g \left(x\right)}{f \left(y\right)}$.

by multiplying by $\mathrm{dx}$ and by $f \left(y\right)$ to separate $x$'s and $y$'s,

$R i g h t a r r o w f \left(y\right) \mathrm{dy} = g \left(x\right) \mathrm{dx}$

by integrating both sides,

$R i g h t a r r o w \int f \left(y\right) \mathrm{dy} = \int g \left(x\right) \mathrm{dx}$,

which gives us the solution expressed implicitly:

$R i g h t a r r o w F \left(y\right) = G \left(x\right) + C$,

where $F$ and $G$ are antiderivatives of $f$ and $g$, respectively.

For an example of a separable equation with an initial condition, please watch this video: