Ordinary differential equations

We summarize here the main results in the theory of ordinary differential equations (ODEs). After recalling some general mathematical definitions, the Cauchy problem in R^n is first considered. Some results are explained without proofs: the Peano’s Ttheorem and the existence and uniqueness of solutions in a rectangle and in a strip. Then, linear ODEs with continuous coefficients are examined, both in the homogeneous and nonhomogeneous case. The generalized Abel’s identity and the Wronskian Theorem are proved. The special case of linear ODEs with constant coefficients is scrutinized. We show that a general ODE of order n can be reduced to the a first-order system of n ODEs. In particular, we scrutinize linear systems and describe the classical method called “separation of variables". Finally, first-order ODEs in Banach spaces are considered. In this framework, the global existence of the Cauchy problem is proved by virtue of the Banach contraction principle.