2 semester


Title of the discipline

Differential equations in applications


Training course, speciality

1, 1-31 81 08 Computer mathematics and systemiс analysis


semester of training



Number of credits



Full name of the lecturer

Amelkin Vladimir Vasilyevich, doctor of physical and mathematical sciences. Professor


The purposes of studying the discipline

The purpose of studying the discipline is to train specialists who have the knowledge and skills to use effectively the basic methods of the theory of differential equations to study the phenomena and processes of reality surrounding us.

Studying the discipline solves the following problem:

– acquiring practical skills in solving mathematical problems, constructing and analyzing mathematical models described by differential equations.

As a result of the training, the undergraduate must


– the basic methods of making up differential models of real phenomena and processes;

– methods for qualitative research of differential models;

– the basic facts of the theory of stability of motion;

     be able to:

 – Practically apply the methods of making up differential models to the construction of specific models from various fields of natural science and technology;

– use methods of a qualitative nature for the study of specific practical problems;

– Solve stability problems by methods of stability theory, both in the case of instantaneous perturbations and in the case of constantly acting perturbations.




Materials of the courses “Algebra and Number Theory”, “Geometry”, “Mathematical Analysis”, “Differential Geometry”, “Differential Equations”.



Content of the


Electrical circuits. Related oscillators. Van der Pol oscillator. Homogeneous perturbations of a harmonic oscillator. The Lienard oscillator of fading out free oscillations. Models of relaxation oscillations. Movement of the gyro in the gimbal suspension. Vibrations of the satellite in the plane of the elliptical orbit. Models with soft and hard modes. Isochronous oscillations in problems of telecommunications. Newton systems with a velocity function that is quadratic in velocity. Models of closed and non-closed systems. R.M.Gudvin economic models of with limit cycles. Model of the conflicts behavior of animals. L.F.Richardson theory of conflict. F.U. Lancaster models of fighting. Model of the detection of diabetes. One problem of the mathematical theory of epidemics. Mathematical model of tumor growth. E.Ziman model of the pulsation of the heart and nerve impulse. Models of road transport.




1.Amelkin, V.V. Differential equations in applications. Izd.3-e / VV Amelkin. – Moscow: The book house “Librocom”, 2008. – 208 p.

2. Ponomarev, K.K. Making up of differential equations / K.K. Ponomarev. – Minsk: The Higher School, 1973. – 560 p.

3.Braun, M. 0Differential Equations and Their Applications. An Introduction to Applied Mathematics. Ed. 3-rd / M.Braun. – New York. Heidelberg. Berlin: Springer – Verlag, 1983. – 546 p.



Methods of teaching

Explanatory-illustrative, reproductive, part-time-search.



Language of learning



Conditions (requirements), routine control

Test works. Reports on laboratory works. Reports on laboratory work with oral protection. The score on the exam is set taking into account: the current assessment – 40%, the oral answer on the exam – 60%.



Form of current


Test, exam