260_pdfsam_math 54 differential equation solutions odd

260_pdfsam_math 54 differential equation solutions odd -...

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Chapter 4 37. Comparing the given homogeneous equations with mass-spring oscillator equation (13) in Section 4.7, [inertia] y 0 +[damping] y 0 + [stifness] y =0 , we see that in equations (a) through (d) the damping coefficient is 0. So, the behavior, oF solutions, as t + , depends on the sign oF the stifness coefficient “ k ”. (a) k ”= t 4 > 0. This implies that all the solutions remain bounded as t + . (b) k ”= t 4 < 0. The stifness oF the system is negative and increases unboundedly as t + . It reinForces the displacement, y ( t ), with magnitude increasing with time. Hence some solutions grow rapidly with time. (c) k ”= y 6 > 0. Similarly to (a), we conclude that all the solutions are bounded. (d) k ”= y 7 . The Function f ( y )= y 7 is positive For positive y and negative iF y is negative. Hence, we can expect that some oF the solutions (say, ones satisFying negative initial conditions) are unbounded.
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Unformatted text preview: (e) k = 3 + sin t . Since | sin t | 1 For any t , we conclude that k 3 + ( 1) = 2 > , and all the solutions are bounded as t + . (f) Here there is positive damping b = t 2 increasing with time, which results an increasing drain oF energy From the system, and positive constant stifness k = 1. Thus all the solutions are bounded. (g) Negative damping b = t 2 increases (in absolute value) with time, which imparts energy to the system instead oF draining it. Note that the stifness k = 1 is also negative. Thus we should expect that some oF the solutions increase unboundedly as t + . 39. IF a weight oF w = 32 lb stretches the spring by ` = 6 in = 0 . 5 Ft, then the spring stifness must be k = w ` = 32 . 5 = 64 (lb / Ft) . 256...
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This note was uploaded on 03/29/2010 for the course MATH 54257 taught by Professor Hald,oh during the Spring '10 term at University of California, Berkeley.

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