Ordinary Differential Equations/Separable equations: Separation of variables

Definition 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 . 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 0 100% #FFFFFF 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 occurring above involving is well-defined. Examples Exercises