Curriculum

Week 1. First-order ordinary differential equations, definition, physical examples. Initial value problems. Existence and uniqueness of the solution (Peano existence theorem and Picard-Lindelöf theorem). Lipschitz condition.

Weeks 2-3. Directly integrable, autonomous and separable differential equations. Application and modeling, readioactive decay, physical eyxamples. The logistic model.

Week 4. Homogeneous equations, reduction to separable form. Linear substitution. Exact differential equations.

Week 5. First order linear differential equations. General solution of nonhomogeneous linear differential equations. Finding a particular solution, variation of parameters. Models of chemical reactions, examples.

Week 6. Higher order linear differential equations. General solution, basis, particular solution. Linearity principle. Existence and uniqueness theorem for initial value problems. Linear independence of solutions, Wronski determinant.

Weeks 7-8. Higher order homogeneous equations with constant coefficients. Characteristic polynomial, characteristic equation (real roots, complex roots, higher multiplicity). General solution of nonhomogeneous linear differential equations. Method of undetermined coefficients. Physical examples.

Week 9. Systems of first order differential equations. Existence and uniqueness theorem. Linear systems. Existence and uniqueness in the linear case, linearity principle. Basis, general solution, Wronskian. Homogeneous systems with constant coefficients. Conversion of an n-th order differential equation to a system.

Week 10. Laplace transforms, definition. Existence theorem for Laplace transforms. Properties of the Laplace transform. Application of the Laplace transform for the solution of higher order linear differential equations and linear systems. Power series method.

Weeks 11-12. Phase portraits fo planar systems, the trace-determinant plane.

Weeks 13-14. Qualitative methods for nolinear systems. Linearization of nonlinear systems. Criteria for stability. Liapunov’s method. First integrals. Liapunov functions. Applications, the Lotka-Volterra population model. Bendixson’s criterion, Bendixson-Dulac theorem.