Ordinary Differential Equations/Separable equations: Separation of variables


A separable ODE is an equation of the form


for some functions  ,  . In this chapter, we shall only be concerned with the case  .

We often write for this ODE


for short, omitting the argument of  .

[Note that the term "separable" comes from the fact that an important class of differential equations has the form


for some  ; hence, a separable ODE is one of these equations, where we can "split" the   as  .]

Informal derivation of the solutionEdit

Using Leibniz' notation for the derivative, we obtain an informal derivation of the solution of separable ODEs, which serves as a good mnemonic.

Let a separable ODE


be given. Using Leibniz notation, it becomes


We now formally multiply both sides by   and divide both sides by   to obtain


Integrating this equation yields




this shall mean that   is a primitive of  . If then   is invertible, we get


where   is a primitive of  ; that is,  , now inserting the variable of   back into the notation.

Now the formulae in this derivation don't actually mean anything; it's only a formal derivation. But below, we will prove that it actually yields the right result.

General solutionEdit

Theorem 2.1:

Let a separable, one-dimensional ODE


be given, where   is never zero. Let   be an antiderivative of   and   an antiderivative of  . If   is invertible, the function


solves the ODE under consideration.


By the inverse and chain rules,


since   is never zero, the fraction occuring above involving   is well-defined.