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Chapter 1 - CHAPTER 1 Introduction to Dierential Equations...

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CHAPTER 1 Introduction to Differential Equations 1.1 Basic Terminology Most of the phenomena studied in the sciences and engineering involve processes that change with time. For example, it is well known that the rate of decay of a radioactive material at time t is proportional to the amount of material present at time t . In mathematical terms this says that dy dt = ky, k a negative constant (1) where y = y ( t ) is the amount of material present at time t . If an object, suspended by a spring, is oscillating up and down, then Newton’s Second Law of Motion ( F = ma ) combined with Hooke’s Law (the restoring force of a spring is proportional to the displacement of the object) results in the equation d 2 y dt 2 + k 2 y = 0 , k a positive constant (2) where y = y ( t ) denotes the position of the object at time t . The basic equation governing the diffusion of heat in a uniform rod of finite length L is given by ∂u ∂t = k 2 2 u ∂x 2 (3) where u = u ( x, t ) is the temperature of the rod at time t at position x on the rod. Each of these equations is an example of what is known as a differential equation. DIFFERENTIAL EQUATION A differential equation is an equation that contains an unknown function together with one or more of its derivatives. Here are some additional examples of differential equations. Example 1. (a) y = x 2 y - y y + 1 . (b) x 2 d 2 y dx 2 - 2 x dy dx + 2 y = 4 x 3 . (c) 2 u ∂x 2 + 2 u ∂y 2 = 0 (Laplace’s equation) (d) d 3 y dx 3 - 4 d 2 y dx 2 + 4 dy dx = 3 e - x . TYPE As suggested by these examples, a differential equation can be classified into one of two general categories determined by the type of unknown function appearing in the equation. If the 1
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unknown function depends on a single independent variable, then the equation is an ordinary differ- ential equation ; if the unknown function depends on more than one independent variable, then the equation is a partial differential equation . According to this classification, the differential equations (1) and (2) are ordinary differential equations, and (3) is a partial differential equation. In Exam- ple 1, equations (a), (b) and (d) are ordinary differential equations and equation (c) is a partial differential equation. Differential equations, both ordinary and partial, are also classified according to the highest- ordered derivative of the unknown function. ORDER The order of a differential equation is the order of the highest derivative of the unknown function appearing in the equation. Equation (1) is a first order equation, and equations (2) and (3) are second order equations. In Example 1, equation (a) is a first order equation, (b) and (c) are second order equations, and equation (d) is a third order equation. In general, the higher the order the more complicated the equation. In Chapter 2 we will consider some first order equations and in Chapter 3 we will study certain kinds of second order equations.
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