By the end of the course, the student will learn the following components of the theory: 

• Domains for the differential equations, integral curves,  trajectories, and phase portraits.
• The definitions of local and global existence and uniqueness of solutions.  Examples of the phenomena in models. 
• Theorems of existence and uniqueness of solutions with various sufficient conditions. The instance of nowhere uniqueness.
• Definitions of continuous and differentiable dependence of solutions on initial data and parameters. 
• Theorems on the dependence of solutions on initial data and parameters. 
• Method of the small parameter in the non-critical case. 
• Analysis of linear homogeneous and inhomogeneous systems of differential equations, focusing on the methods of solutions, description of individual and collective behaviour. Spaces of solutions.
• Periodic systems of linear equations, criteria of existence for periodic solutions.
• Higher-order linear equations. Connection with the first-order linear systems. The initial value problem.
• Definitions of Lyapunov stability. Linear, quasilinear systems. Stability by linearization. 
• Lyapunov’s second method. Application to mechanical and electrical models.