5397-Chap1

5397-Chap1 - Part 3: DIFFERENTIAL EQUATIONS Chapter 1...

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Part 3: DIFFERENTIAL EQUATIONS

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Chapter 1 Introduction and 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 ±nite 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 which contains an unknown function together with one or more of its derivatives. Here are some additional examples of diﬀerential equations. 1
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 classifed into one oF two general categories determined by the type oF unknown Function appearing in the equation. IF the unknown Function depends on a single independent variable, then the equation is an ordinary diﬀerential equation ; iF the unknown Function depends on more than one independent variable, then the equation is a partial diﬀerential equation . The diﬀerential equations (1) and (2) are ordinary diﬀerential equations, and (3) is a partial diﬀerential equation. In Example 1, equations (a), (b) and (d) are ordinary diﬀerential equations and equation (c) is a partial diﬀerential equation. 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 frst order equation, and equations (2) and (3) are second order equa- tions. In Example 1, equation (a) is a frst order equation, (b) and (c) are second order equations, and equation (d) is a third order equation. The obvious question that we want to consider is that oF “solving” a given diﬀerential equation. SOLUTION A solution of a diﬀerential equation is a Function defned on some interval I (in the case oF an ordinary diﬀerential equation) or on some domain D in two or higher dimensional space (in the case oF a partial diﬀerential equation) with the property that the equation reduces to an identity when the Function is substituted into the equation.

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5397-Chap1 - Part 3: DIFFERENTIAL EQUATIONS Chapter 1...

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