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# sol2 - ACM 95b/100b Solutions for Problem Set 2 Sean Mauch...

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ACM 95b/100b Solutions for Problem Set 2 Sean Mauch [email protected] January 18, 2002 1 Problem 1 Problem. Four ladybugs start at the corners of a unit square ( ± 1 2 ± ı 2 ). At t 0 , each beetle moves along a smooth trajectory that is continuously adjusted to point toward the next beetle along the square in a counterclockwise direction. At any given time, all four beetles are moving at the same speed. 1. Using complex dependent variables to simplify your formulation, write down a system of four differential equations with initial conditions that governs the motion of the beetles as they spiral in to meet at the origin. 2. Use the symmetry of the problem to replace this system with a single first order initial value problem that describes the motion of one of the beetles along its trajectory. 3. Solve this IVP and sketch the solution. How many times does the beetle spiral around the origin? 4. Determine the length of this trajectory from t 0 until the time at which the beetles meet at the origin. Solution. 1. Let z k be the position of the beetle that is initially in quadrant k + 1. We write down the system of differential equations that govern their motion. We choose a speed that simplifies the equations namely, the speed is the distance to the adjacent beetle. z 0 0 = z 1 - z 0 z 0 1 = z 2 - z 0 z 0 2 = z 3 - z 2 z 0 3 = z 0 - z 3 The initial conditions are z k ( t 0 ) = e ı (2 k +1) π/ 4 2 . 2. We note that the beetles are each separated by an angle of π/ 2. The position of beetle k + 1 mod 4 is ı times the position of beetle k . This observation allows us to reduce the problem to a single initial value problem for beetle zero. We take the start time to be t 0 = 0. z 0 = ız - z, z (0) = 1 + ı 2 The position of beetle k is z k = ı k z . 1

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3. We solve this initial value problem. The general solution of the differential equation is z = c e ( ı - 1) t . We determine the constant of integration from the initial condition. z = e ( ı - 1) t + ıπ/ 4 2 The angle of the beetle is t + π/ 4. Its radius is e - t . Thus the beetle spirals around the origin an infinite number of times. The four beetle paths are shown in Figure 1. Figure 1: The paths of the four beetles. 4. We integrate to determine the length of the trajectory. L = Z 0 | z 0 | d t = Z 0 ( ı - 1) e ( ı - 1) t + ıπ/ 4 2 d t = Z 0 e - t d t = 1 The trajectory has unit length. 2 Problem 2 Problem. For the following linear 2 nd order non-homogeneous ODE’s, state the intervals in which a unique solution satisfying initial conditions y ( x 0 ) = y 0 , y 0 ( x 0 ) = y 0 0 is guaranteed to exist, where x 0 is any point in the interval.
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