The numerical solution of ordinary differential equations

Introduction
Terminology, classification of differential equations (DE) and their systems.
Initial value problems (IVP).
Transformation of higher order DE into systems of first order DE.
Existence and uniqueness of IVP solution.
Lipschitz condition, conditions expressed via partial derivatives.
IVP preconditionality
Numerical methods for IVP solution.
Principles of IVP solution methods.
Euler method.
Method order.
Approximation errors.
Method convergence.
Method order, Euler method order and global error.
Round errors influence and error estimation by half-step method.
One-step methods.
Taylor methods.
Runge-Kutta methods, error estimation.
Multi-step methods.
Linear k-step method.
Discretization error.
Overview of some multi-step methods.
Solution stability, choice of IVP solution method