Ordinary Differential Equations/Separable equations: Separation of variables

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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 solution[edit]

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 solution[edit]

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.