Syllabus
1. Peano Theorem on local existence of solutions, local and global uniqueness, sufficient conditions for local uniqueness. Maximal solution - existence, characterisation. Gronwall lemma. Continuous and differentiable dependence of solutions on parameters or initial value.
2. Linear equations: global existence and uniquness. Fundamental matrix, Wronskian, Liouville's formula. Variation of parameters in integral form. Linear systems with constant coefficients. Exponential of a matrix and its properties. Stable, unstable and central subspaces.
3. Stability, asymptotic stability. Uniform stability. Stability of linear equations. Linearized stability and unstability.
4. First integral, orbital derivative. Existence of first integrals. Application: method of characteristics.
5. Higher order equations: reformulation as a first order system. Theorems on local existence and uniquness. Variation of parameters.
6. Stability II: Lyapunov function, theorems on stability and asymptotic stability. Ljapunov equation.
7. Floquet theory: logaritm of a matrix. Existence of periodic solutions and their stability.
Annotation
A course for bachelor's program in General Mathematics.
Recommended for specializations Mathematical Analysis and Mathematical Modelling and Numerical Analysis.