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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
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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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, if e6t u - u - -8° Such a value of t will exist if u is such that F(u) > 1, where F FIGURE 5.1 The function F(u). и F = - 2 u - . -8° น The graph of F is shown in Figure 5.1. The condition F > 1 is satisfied if u > 8, but not otherwise. Hence the particle will pass through the origin if u > 8.
Transcribed Image Text:that is, if e6t u - u - -8° Such a value of t will exist if u is such that F(u) > 1, where F FIGURE 5.1 The function F(u). и F = - 2 u - . -8° น The graph of F is shown in Figure 5.1. The condition F > 1 is satisfied if u > 8, but not otherwise. Hence the particle will pass through the origin if u > 8.
Problem 5.4 An overdamped harmonic oscillator satisfies the equation x+10x + 16x = : 0. At time t = O the particle is projected from the point x = 1 towards the origin with speed u. Find x in the subsequent motion. Show that the particle will reach the origin at some later time t if и 2 = e e6t - и 8 How large must u be so that the particle will pass through the origin? Solution The equation of motion is solved in the standard manner by seeking solutions of the form x = eat. Then λ must satisfy the equation the roots of which are λ == 22+102+16 = 0, -2, -8. We have thus found the pair of solutions x = -2t e-8t The general solution of the equation of motion is therefore x = Ae -2t + Be-8t ' where A and B are arbitrary constants. The initial conditions x = 1 and x = -u when t = 0 give the equations A + B = 1, 2A + 8B = u, from which it follows that A = particle is therefore given by - — (u – 8), B = (u 2). The motion of the x = 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,
Transcribed Image Text:Problem 5.4 An overdamped harmonic oscillator satisfies the equation x+10x + 16x = : 0. At time t = O the particle is projected from the point x = 1 towards the origin with speed u. Find x in the subsequent motion. Show that the particle will reach the origin at some later time t if и 2 = e e6t - и 8 How large must u be so that the particle will pass through the origin? Solution The equation of motion is solved in the standard manner by seeking solutions of the form x = eat. Then λ must satisfy the equation the roots of which are λ == 22+102+16 = 0, -2, -8. We have thus found the pair of solutions x = -2t e-8t The general solution of the equation of motion is therefore x = Ae -2t + Be-8t ' where A and B are arbitrary constants. The initial conditions x = 1 and x = -u when t = 0 give the equations A + B = 1, 2A + 8B = u, from which it follows that A = particle is therefore given by - — (u – 8), B = (u 2). The motion of the x = 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,
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