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# S11 - CHAPTER 1 Introduction to Differential Equations 1.1...

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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 .

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