tailieunhanh - Ebook A first course in differential equations (3rd edition): Part 2

(BQ) Part 2 book "A first course in differential equations" has contents: Linear systems (matrices and linear systems, the eigenvalue problem, phase plane analysis, nonhomogeneous systems), nonlinear systems, computation of solutions. | 4 Linear Systems The first three chapters dealt a single first-order or second-order linear differential equation. A natural next step is to examine coupled systems of differential equations with several unknown functions. Many problems from the pure and applied sciences have multiple components linked together in some manner, and it is natural to formulate those as first-order systems. For example, two chemical reactors may be coupled together with two concentrations to account for, one in each reactor. Or an electrical circuit may have coupled loops, each carrying a different current. Predator-prey models in ecology involve two animal species, the prey and predator, and their populations are coupled together through their interactions. In this text we consider only planar, or two-dimensional systems, consisting of two first-order equations in two unknown functions x(t) and y(t). As we will see, second-order equations, studied in Chapter 2, are equivalent to such a planar system, but a systems approach exposes an insightful geometrical structure revealing the dynamics. Our goal is to show what to expect from a system, including general solutions and how to display them graphically for easy visualization. This is carried out by introducing matrix notation, a language that greatly simplifies their representation and which extends in a straightforward way to systems of higher dimensions. 178 4. Linear Systems Linear Systems vs. Second-Order Equations Consider the damped oscillator equation mx′′ + γx′ + kx = 0 with unknown displacement x = x(t). If we introduce the velocity y = y(t) as another dependent function, then x′ = y and the damped oscillator equation becomes my ′ + γy + kx = 0, or, my ′ = −kx − γy. Therefore, x′ = y, k y′ = − m x − γ m y, This is a system of two first-order equations in two unknowns x = x(t) and y = y(t) and it is completely equivalent to the second-order damped oscillator equation. Remark We can always reduce a second-order linear .