anonymous
  • anonymous
Consider the overdamped harmonic oscillator d^2y/dt^2 + 5dy/dt+ 2y = 0 Show that for every solution y(t) to this differential equation, there is at most one value of t such that y(t) = 0 (i.e. none of the solutions oscillate).
Mathematics
  • Stacey Warren - Expert brainly.com
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chestercat
  • chestercat
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anonymous
  • anonymous
@TuringTest
TuringTest
  • TuringTest
I'm not sure I understand In this situation of over-damping we have that \(y(t)\to0\) as \(t\to\infty\), so it seems you are asking to prove something that is not quite true... there is no finite value t at which y(t)=0
anonymous
  • anonymous
well that would explain why this isn't making sense.. is there a calculation that would prove that this is the casE?

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TuringTest
  • TuringTest
check that the roots of the characteristic polynomial are both negative, so you have something of the form\[\large y(t)=c_1e^{r_1t}+c_2e^{r_2t}\]where\[r_1,r_2<0\]that means that \(y\to0\) as \(t\to\infty\)
anonymous
  • anonymous
ya they are both negative
TuringTest
  • TuringTest
you can prove, in general, that if the discriminant of the characteristic polynomial is >0, the roots will both be negative a proof of that in general can be found here: http://tutorial.math.lamar.edu/Classes/DE/Vibrations.aspx look at part 2 under "Free, Damped Vibrations"
anonymous
  • anonymous
thank you so so so much!!!
TuringTest
  • TuringTest
welcome :)

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