The purpose of this talk to discuss some methods that are used in non-linear differential equations. I will work through the framework for dealing with some Non-linear ODEs, discussing: local existence, uniqueness and global existence. The techniques used will be that of: contraction maps, Gronwall’s lemma, energy methods and bootstrapping. This framework serves as a basis for the analysis PDEs (far more complicated equations). The talk aims to highlight in detail the concept of bootstrapping. Bootstrapping is a novel way of proving boundedness out of seemingly nothing and can be thought of as a continuous extension of proof by induction. It is an invaluable tool used heavily in the analysis of dispersive PDEs.