Systems of Differential Equations and Models in Physics, Engineering and Economics

Coordinating professor: Valeriu Prepelita

Bucharest,

July, 2010

Table of Contents

1.Importance and uses of differential equations4

1.1.Creating useful models using differential equations4

1.2.Real-life uses of differential equations5

2.Introduction to differential equations6

2.1.First order equations6

2.1.1.Homogeneous equations6

2.1.2.Exact equations8

2.2.Second order linear equations10

3.Systems of differential equations14

3.1.Systems of linear differential equations16

3.1.1.Systems of linear differential equations with constant coefficients22

3.2.Systems of first order equations27

3.2.1.General remarks on systems27

3.2.2.Linear systems. Case n=231

3.2.3.Nonlinear systems. Volterra’s Prey-predator equations38

3.3.Critical points and stability for linear systems44

3.3.1.Bounded input bounded output stability44

3.3.2.Critical points44

3.3.3.Methods of determination of stability of linear systems56

3.4.Simple critical points on nonlinear systems60

3.5.Nonlinear mechanics. Conservative systems.66

4.Applications of differential equations73

4.1.Growth, decay and chemical reactions73

4.2.Free fall75

4.3.Retarded fall77

5.Applications of systems of differential equations78

5.1.Passive L-R-C circuits having multiple loops78

5.2.The predator/prey model83

5.3.Solving the structural dynamics equations for periodic applied forces84

5.3.1.Vibrating vertically suspended cable87

5.4.Example of induced oscillations to a pendulum attached to a string92

5.5.Earthquake simulation for a pile embedded in soil96

6.Conclusions102

6.1.Personal contributions and notes103

7.Appendix104

7.1.Vibrating vertically suspended cable104

7.2.Example of induced oscillations to a pendulum attached to a string109

7.3.Earthquake simulation for a pile embedded in soil112

8.References116

1. Importance and uses of differential equations

So far as the theories of mathematics are about reality, they are not certain; so far as they are certain, they are not about reality. Albert Einstein (1879-1955) The subject of differential equations is large, diverse and useful. Differential equations can be studied on their own or they can be studied by different scientists, whether they are physicists, engineers, biologists or economists. Many physical and even abstract systems can be explained by transposition into this mathematical concept.

2.1. Creating useful models using differential equations The broad area of applied mathematics usually consists in:

* Formulation of a mathematical model to describe a physical situation * Precise statement and analysis of an appropriate mathematical model * Approximate numerical calculation of important physical quantities * Comparison of physical quantities with experimental data to check the validity of the model. Although tasks are never clear enough, physicists and engineers handle parts 1 and 4 (formulation of mathematical model and comparison of physical quantities), while parts 2 and 3 (precise statement and approximate numerical calculation) are directed to mathematicians. In order for these four steps to be followed in the most efficient manner, when a mathematical model is formulated, one must take into account both the lack of precision when describing a physical situation and the inability to analyze the mathematical model which may be forthcoming. Since nature is very complex and changes may occur unexpectedly, the mathematician cannot argue that a solution exists and it is unique because the physical situation seems to prove so, since by the time the problem reaches him, the description is no longer accurate. It becomes an approximate model. Therefore, the mathematician...