Chapters 1-2 - textbook

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C H A P T E R 1 First-Order Differential Equations Among all 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 1.1 How Differential Equations Arise In this section we will introduce the idea of a differential equation through the mathe- matical 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 , the 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 m dv dt = F, (1.1.1) where v(t) denotes the velocity of the object at time t . We let y(t) denote the displacement of the object at time t . Then, using the fact that velocity and displacement are related via v = dy dt , we can write (1.1.1) as m d 2 y dt 2 = F. (1.1.2) This is an example of a differential equation , so called because it involves derivatives of the unknown function y(t) . 1
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2 CHAPTER 1 First-Order Differential Equations mg Positive y -direction Figure 1.1.1: 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 1.1.1). 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 (1.1.2) that the motion of the object is governed by the differential equation m d 2 y dt 2 = mg, (1.1.3) or equivalently, d 2 y dt 2 = g. Since g is a (positive) constant, we can integrate this equation to determine y(t) . Per- forming one integration yields dy dt = gt + c 1 , where c 1 is an arbitrary integration constant. Integrating once more with respect to t, we obtain y(t) = 1 2 gt 2 + c 1 t + c 2 , (1.1.4) where c 2 is a second integration constant. We see that the differential equation has an infinite number of solutions parameterized by the constants c 1 and c 2 . In order to uniquely specify the motion, we must augment the differential equation with initial conditions that specify the initial position and initial velocity of the object. For example, if the object is released at t = 0 from y = y 0 with a velocity v 0 , then, in addition to the differential equation, we have the initial conditions y( 0 ) = y 0 , dy dt ( 0 ) = v 0 . (1.1.5) These conditions must be imposed on the solution (1.1.4) in order to determine the values of c 1 and c 2 that correspond to the particular problem under investigation. Setting t = 0 in (1.1.4) and using the first initial condition from (1.1.5), we find that y 0 = c 2 .
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