Separable differential equation

Definition

Form of the differential equation

The term separable is used for a first-order differential equation that, up to basic algebraic manipulation, is of the form:

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

Solution method and formula: general solution

It can be solved by rearranging and integrating:

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

It suffices to have just one freely floating additive constant in the answer because the additive constants coming from the two integrals can be merged into one.

In general, the solution to this is in the form of an implicit function rather than an explicit description of $y$ as a function of $x$.

In addition to the above formula of general solutions, it is also possible that there exist additional solutions that are singular solutions. These are solutions of the form:

$y=k,\text{where }g(k)=0$

This family of solutions is usually a discrete, often finite, family of solutions.

Particular cases

Where the derivative depends only on the dependent variable

This is an example of an autonomous differential equation (usually, $x$ is replaced by the letter $t$ denoting time):

$\frac{dy}{dx}=g(y)$

Here, we get:

$\int \frac{dy}{g(y)}=\int 1dx$

Note that performing the integration expresses $x$ in terms of $y$. We need to then do algebraic manipulation to express $y$ explicitly in terms of $x$.

Again, we need to take care of additional solutions of the form:

$y=k,\text{where }g(k)=0$

Where the derivative depends only on the independent variable

This is a situation where the function depends only on $x$:

$\frac{dy}{dx}=f(x)$

We get:

$y=\int f(x)dx$

This is a straightforward explicit functional description.