Systems of Differential Equations and Models in Physics, Engineering and Economics
Universitatea Politehnica Bucuresti - FILS
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 equations 4 1.1. Creating useful models using differential equations 4 1.2. Real-life uses of differential equations 5 2. Introduction to differential equations 6 2.1. First order equations 6 2.1.1. Homogeneous equations 6 2.1.2. Exact equations 8 2.2. Second order linear equations 10 3. Systems of differential equations 14 3.1. Systems of linear differential equations 16 3.1.1. Systems of linear differential equations with constant coefficients 22 3.2. Systems of first order equations 27 3.2.1. General remarks on systems 27 3.2.2. Linear systems. Case n=2 31 3.2.3. Nonlinear systems. Volterra’s Prey-predator equations 38 3.3. Critical points and stability for linear systems 44 3.3.1. Bounded input bounded output stability 44 3.3.2. Critical points 44 3.3.3. Methods of determination of stability of linear systems 56 3.4. Simple critical points on nonlinear systems 60 3.5. Nonlinear mechanics. Conservative systems. 66 4. Applications of differential equations 73 4.1. Growth, decay and chemical reactions 73 4.2. Free fall 75 4.3. Retarded fall 77 5. Applications of systems of differential equations 78 5.1. Passive L-R-C circuits having multiple loops 78 5.2. The predator/prey model 83 5.3. Solving the structural dynamics equations for periodic applied forces 84 5.3.1. Vibrating vertically suspended cable 87 5.4. Example of induced oscillations to a pendulum attached to a string 92 5.5. Earthquake simulation for a pile embedded in soil 96 6. Conclusions 102 6.1. Personal contributions and notes 103 7. Appendix 104 7.1. Vibrating vertically suspended cable 104 7.2. Example of induced oscillations to a pendulum
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 equations 4 1.1. Creating useful models using differential equations 4 1.2. Real-life uses of differential equations 5 2. Introduction to differential equations 6 2.1. First order equations 6 2.1.1. Homogeneous equations 6 2.1.2. Exact equations 8 2.2. Second order linear equations 10 3. Systems of differential equations 14 3.1. Systems of linear differential equations 16 3.1.1. Systems of linear differential equations with constant coefficients 22 3.2. Systems of first order equations 27 3.2.1. General remarks on systems 27 3.2.2. Linear systems. Case n=2 31 3.2.3. Nonlinear systems. Volterra’s Prey-predator equations 38 3.3. Critical points and stability for linear systems 44 3.3.1. Bounded input bounded output stability 44 3.3.2. Critical points 44 3.3.3. Methods of determination of stability of linear systems 56 3.4. Simple critical points on nonlinear systems 60 3.5. Nonlinear mechanics. Conservative systems. 66 4. Applications of differential equations 73 4.1. Growth, decay and chemical reactions 73 4.2. Free fall 75 4.3. Retarded fall 77 5. Applications of systems of differential equations 78 5.1. Passive L-R-C circuits having multiple loops 78 5.2. The predator/prey model 83 5.3. Solving the structural dynamics equations for periodic applied forces 84 5.3.1. Vibrating vertically suspended cable 87 5.4. Example of induced oscillations to a pendulum attached to a string 92 5.5. Earthquake simulation for a pile embedded in soil 96 6. Conclusions 102 6.1. Personal contributions and notes 103 7. Appendix 104 7.1. Vibrating vertically suspended cable 104 7.2. Example of induced oscillations to a pendulum
References: George F. Simmons, Differential Equations with applications and historical notes, McGraw Hill, USA, 1972 Knut Sydsaeter, Peter Hammond, Atle Seierstad, Arne Strom, Further mathematics for economic analysis – second edition, Financial Times , England, 2008 Ronald Shone, Economic dynamics: phase diagrams and their economic application, Cambridge University Press, UK, 2002 Narain Kapur, Mathematical modeling, New Age International Publishers, New Delhi, 1998 Lawrence Perko, Differential equations and dynamical systems, Springer, USA, 2000 E.L D. Polyanin, V. F. Zaitsev, and A. Moussiaux, Handbook of First Order Partial Differential Equations, Taylor & Francis, London, 2002 Allan Jeffrey, Linear algebra and ordinary differential equations, CRC Press, USA, 2007 Aristide Halanay, Differential equations: stability, oscillations, time lags, Elsevier, New York, 1966 -------------------------------------------- [ 5 ]. Vito Volterra (3 May 1860 – 11 October 1940) was an Italian mathematician and physicist, known for his contributions to mathematical biology and integral equations.