Ordinary differential equations I

mathematics

obligatory courses

(in Polish) Oczekuje się dobrej znajomości zagadnień ujętych w sylabusach przedmiotów Analiza matematyczna I.2 oraz Analiza matematyczna II.1.

The lecture presents basic informations on existence, uniqueness and properties of ODE solutions. Elements of the qualitative analysis of solutions are also included. A number of important applications of ODE is discussed.

Ordinary differential equations and their solutions (definition, examples). Initial value problem. Equations of higher order. Solution methods for scalar equations: with separable variables, linear equations, Bernoulli eq., complete differentials.

Local existence and uniqueness. Picard-Lindelof theorem. Dependence on parameters and initial values. Prolongation of solutions.

First order linear systems. The space of solutions. Wronski determinant and Liouville theorem. Systems with constant coefficients. Linear equations of higher order.

Autonomous equations and flows. Vector fields. Liapunov stability and asymptotic stability. Phase space and phase curves. Phase curves for a 2 dimentional linear system. The pendulum. Liapunov stability of solutions. The logistic model. The Lotka-Volterra system of equations.

1. Arrowsmith D.K., Place C.M. - Ordinary Differential Equations, Approach

with Applications, Chapman & Hall.

2. Hirsch M.W., Smale S. - Differential Equations, Dynamical Systems and

Linear Algebra, Academic Press.

Knowledge and skills:

The students:

1. know the concepts of differential equation, the solutions of initial value problem (IVP), can verify whether the specified function is the solution of ODE or IVP;

2. can solve: separable, homogeneous, Bernoulli ODEs;

3. know the sufficient conditions of existence and uniqueness of solution of IVP;

4. can give an example of IVP with infinite number of solutions;

know the theorem about extending solutions of ODEs and can give an example of IVP which cannot be extended beyond some finite interval;

5. can solve the linear ODEs;

6. can convert higher order ODE to a system of the first order ODEs;

7. can find the fundamental matrices for systems of linear ODEs;

8. know the concept of vector field;

9. know the concept of equilibrium points and know the definitions of asymptotic and Lyapunov stabilities of equilibrium points;

10. can veryfy the stability of an equilibrium point;

11. knows examples of applications of ODEs in sciences and real life.

Competence:

The students understand the role of ODEs in modelling natural processes.