1. SylabUZ
2. Faculty of Exact and Natural Sciences
3. winter term 2022/2023
4. Mathematics - First-cycle studies leading to Bachelor's degree
5. Differential Equations

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Differential Equations - course description

Aim of the course

The main aim of this course is to familiarize students with the basic theory of ordinary differential equations: finding solutions of first-order and second-order ODE as well as first-order systems of ODE, the existence and the uniqueness of solutions of ODE, testing stability of critical points and making phase portraits of linear system in the plane.

Prerequisites

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Mathematical Analysis 1 and 2, Linear Algebra 1 and 2.

Scope

Lecture

1. Basic concepts: n-order ordinary differential equation, system of n-order ordinary differential equations, solution of ODE, continuation of solution, saturated solutions, general and particular solution, integral curves, first integrals, phase space. (2 hours)
2. First-order ordinary differential equations. Examples of phenomena leading to ODE. Geometrical interpretation of ODE. Separable equations and the types of equations reducible to separable equations. Linear ODE and equations reducible to linear equations (Bernoulli equations, Riccati equations). Exact equations. (5 hours)
3. Existence and uniqueness of local solutions of initial problems for ODE. Cauchy problem for ODE. Picard-Lindelöf theorem. Gronwall's lemma. Peano theorem. Extension of solutions of the initial problems for ODE. Dependence of the solution to Cauchy problem on initial conditions and the right-hand side of the equation. (6 hours)
4. Second-order ordinary differential equations. Physical motivation. Types of equations reducible to first-order ordinary differential equations. Linear second-order differential equations. (4 hours)
5. Systems of linear first-order differential equations. Existence and uniqueness of solutions. Systems of homogeneous equations, fundamental matrix. Systems of homogeneous equations with constant coefficients. Systems of inhomogenous equations. (6 hours)
6. Elements of the qualitative theory of ODE. Classification and stability of critical points of systems of linear ODE in the plane. Phase portraits of linear systems. Classification and stability of critical points of systems of nonlinear ODE in the plane. Lyapunov stability. Lyapunov function and fundamental stability theorems. (7 hours)

 

Classes
1. Solving first-order ordinary differential equations: separable equations and the types of equations reducible to separable equations, linear equations, Bernoulli equations, Riccati equations, exact equations. Solving exercises related to physical phenomena which should be described in terms of Cauchy problems for ODE. (8 hours)
2. Solving exercises with the use of existence and uniqueness theorems of local solutions of initial problems for ODE. (3 hours)
3. Test. (2 hours)
4. Solving second-order ordinary differential equations by reducing them to first-order ordinary differential equations. Solving second-order linear ordinary differential equations. (5 hours)
5. Solving systems of first-order linear ordinary differential equations - computation of fundamental matrix. (5 hours)
6. Examination of stability of critical points of systems of first-order linear ODE. Sketching phase portraits. (5 hours)
7. Test. (2 hours)

Teaching methods

Traditional lectures; classes with the lists of exercises to solve by students.

Learning outcomes and methods of theirs verification

Outcome description	Outcome symbols	Methods of verification	The class form

Assignment conditions

Classes: learning outcomes will be verified through homeworks and two tests consisted of exercises of different degree of difficulty. A grade determined by the sum of points from these homeworks and test is a basis of assessment.
Lecture: final exam. A grade determined by the sum of points from that exam is a basis of assessment.
A grade from the course is consisted of the grade from classes (50%) and the grade from the final exam (50%). To take a final exam, students must receive a positive grade from classes. To attain a pass in the course students are required to pass the final exam.

Recommended reading

1. M. Braun, Differential Equations and Their Applications, An Introduction to Applied Mathematics, Springer, New York 1983.
2. William E. Boyce, Richard C. DiPrima, Elementary differential equations and boundary value problems, Wiley, New York 2001.
3. Arrowsmith D.K., Place C.M.: Ordinary Differential Equations, Approach with Applications, Chapman & Hall, London 1982.

 

Further reading

1. Hartman P.: Ordinary Differential Equations, Wiley & Sons, New York 1964.

Notes

Modified by dr Ewa Sylwestrzak-Maślanka (last modification: 16-12-2023 22:08)