tailieunhanh - Ebook Differential equations and linear algebra (3th edition): Part 1

(BQ) Part 1 book "Differential equations and linear algebra" has contents: First order equations, second order equations, graphical and numerical methods, linear equations and inverse matrices. | DIFFERENTIAL EQUATIONS AND LINEAR ALGEBRA THIRD HDIÌION STEPHEN w. GOODE SCOTT A. ANNIN First-Order Differential Equations Among oil of the mathematical disciplines the theory of differential equations is the most important. It furnishes the explanation of all those elementary manifestations of nature which involve time. Sophus Lie How Differential Equations Arise In this section we will inttoduce the idea of a differential equation through the mathematical formulation of a variety of problems. We then use these problems throughout the chapter to illustrate the applicability of the techniques introduced. Newton s Second Law of Motion Newton s second law of motion states that for an object of constant mass m die sum of the applied forces acting on the object is equal to the mass of the object multiplied by the acceleration of the object. If the object is moving in one dimension under the influence of a force F then the mathematical statement of this law is F. at where v denotes the velocity of the object al time f. We let y r denote the displacement of the object at time t. Then using the fact that velocity and displacement are related via _ dy dt we can write as 1 7 F. dt2 This is oil cxiuiiplc 01 a dilfcrriitial irijuiiihiii so Called because it involves dêrivũỉivês of the unknown function y r . 1 CHAPTER 1 First-Order Differential Equations Positive y-direction Figure . Object falling under the influence of gravity. Gravitational Force As a specific example consider the case of an object falling freely under the influence of gravity see Figure . In this case the only force acting on the object is F mg. where g denotes the constant acceleration due to gravity. Choosing the positive y-direction as downward it follows from Equation that the motion of the object is governed by the differential equation d2y _ mg. at or equivalently. o dt2 s Since g is a positive constant we can integrate this equation to determine y r . .