Chapter 1 Geometric Approach to Differential Equations In a basic elementary differential equations course, the emphasis is on linear dif- ferential equations. One example treated is the linear second-order differential equation (1.1) m ¨ + b ˙ + k x = 0, where we write ˙ for dx dt and ¨ for d2x dt2 , and m, k 0 and b ≥ 0. This equation is a model for a linear spring with friction and is also called a damped harmonic oscillator. To determine the future motion, we need to know both the current position x and the current velocity ˙. Since these two quantities determine the motion, it is natural to use them for the coordinates writing v = ˙, we can rewrite this equation as a linear system of first-order differential equations (involving only first derivatives), so m ˙ = m ¨ = −k x − b v: ˙ = v, (1.2) ˙ = − k m x − b m v. In matrix notation, this becomes ˙ ˙ = 0 1 − k m − b m x v . For a linear differential equation such as (1.1), the usual solution method is to seek solutions of the form x(t) = eλ t , for which ˙ = λ eλ t and ¨ = λ2 eλ t . We need m λ2 eλ t + b λ eλ t + k eλ t = 0, 3
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