DIFFERENTIAL EQUATIONS (MATHEMATICS III)

General Foundation

The course is also offered to exchange programme students.

By the end of the course students will be able to: Model a simple physical system to obtain a first order differential equation. Test the plausibility of a solution to a differential equation (DE) which models a physical situation by using reality-check methods such as physical reasoning, looking at the graph of the solution, testing extreme cases, and checking units. Visualize solutions using direction fields and approximate them using Euler's method. Find and classify the critical points of a first order autonomous equation and use them to describe the qualitative behavior and, in particular, the stability of the solutions. The main equations studied in the course are driven first and second order constant coefficient linear ordinary differential equations and 2x2 systems. For these equations students will be able to: Use known DE types to model and understand situations involving exponential growth or decay and second order physical systems such as driven spring-mass systems or LRC circuits. Solve the main equations with various input functions including zero, constants, exponentials, sinusoids, step functions, impulses, and superpositions of these functions. Understand and use fluently the following features of the linear system response: solution, stability, transient, steady-state, amplitude response, phase response, amplitude-phase form, weight and transfer functions, pole diagrams, resonance and practical resonance, fundamental matrix. Use the following techniques to solve the differential equations described above: characteristic equation, exponential response formula, Laplace transform, convolution integrals, Fourier series, complex arithmetic, variation of parameters, elimination and anti-elimination, matrix eigenvalue method. Understand the basic notions of linearity, superposition, and existence and uniqueness of solutions to DE's, and use these concepts in solving linear DE's. Draw and interpret the phase portrait for autonomous 2x2 linear constant coefficient systems. Linearize an autonomous non-linear 2x2 system around its critical points and use this to sketch its phase portrait and, in particular, the stability behavior of the system.

Differential Equations: Definition and Properties. DE 1st and higher order linear and nonlinear Systems of Differential Equations. Laplace Transform and Fourier series. Partial Differential Equations, Separation of variables. Boundary Value Problems

09-06-2020