Ordinary Differential Equations/Separable equations: Separation of variables

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Definition[edit]

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

.

Define

;

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.

Proof:

By the inverse and chain rules,

;

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

Examples[edit]

Exercises[edit]