Equivalence of third-order ordinary differential equations to Chazy equations I-XIII

In the paper we solve the equivalence problem of the third-order ordinary differential equations quadratic in the second-order derivative. For this class of equations the invariants of the group of point equivalence transformations and the invariant differentiation operators are constructed. Using these results the invariants of 13 Chazy equations were calculated. We provide examples of finding equivalent equations by use of their invariants. Also two new examples of the equations linearizable by a local transformation are found. These are a particular case of Chazy-XII equation and a Schwarzian equation.