I'm having difficulty comprehending the response in this solution, regarding the question, 'How large must u be so that the particle will pass through the origin?' Could you please clarify how the solution justifies this aspect? that is, ife6tu -u --8°Such a value of t will exist if u is such that F(u) > 1, whereFFIGURE 5.1 The function F(u).иF=-2u -.-8°นThe graph of F is shown in Figure 5.1. The condition F > 1 is satisfied ifu > 8, but not otherwise. Hence the particle will pass through the origin if u > 8. Problem 5.4An overdamped harmonic oscillator satisfies the equationx+10x + 16x = : 0.At time t = O the particle is projected from the point x = 1 towards the origin withspeed u. Find x in the subsequent motion.Show that the particle will reach the origin at some later time t ifи2= ee6t-и 8How large must u be so that the particle will pass through the origin?SolutionThe equation of motion is solved in the standard manner by seeking solutions of theform x = eat. Then λ must satisfy the equationthe roots of which are λ==22+102+16 = 0,-2, -8. We have thus found the pair of solutionsx =-2te-8tThe general solution of the equation of motion is thereforex = Ae-2t+ Be-8t'where A and B are arbitrary constants. The initial conditions x = 1 and x = -uwhen t = 0 give the equationsA + B = 1,2A + 8B = u,from which it follows that A =particle is therefore given by-— (u – 8), B=(u 2). The motion of thex = ½ (u − 2)e¯8 – ½ (u — 8)e¯² ¸The particle is at the origin at time t if-(u− 2)e8t — (u - 8)e¯²t = 0,

Domain and Range:-The domain of logarithmic function y=logx is x>0.So the value of y can not…

I'm having difficulty comprehending the response in this solution, regarding the question, 'How large must u be so that the particle will pass through the origin?' Could you please clarify how the solution justifies this aspect?

I'm having difficulty comprehending the response in this solution, regarding the question, 'How large must u be so that the particle will pass through the origin?' Could you please clarify how the solution justifies this aspect?

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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