Ordinary Differential Equations/Separable equations: Separation of variables

DefinitionEdit

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

 .

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 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.

Proof:

By the inverse and chain rules,

 ;

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

ExamplesEdit

ExercisesEdit