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
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.
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.