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Question

Math 6410 • ODE

Spring 2009 • Question 4

Question

Math 6410 • ODE Spring 2009 • Question 4

Consider the scalar equation \[ \ddot {u}+\epsilon (u^2-1)\dot {u}+u=0,\qquad 0<\epsilon \ll 1. \] Apply the method of multiple scales, with variables \(t\) and \(\eta =\epsilon t\) to show that if \(u(0)=a>0\) and \(\dot {u}(0)=0\) then for \(t=O(\epsilon ^{-1})\), \[ u=\frac {2\cos t}{\sqrt {1+\left (\frac {4}{a^2}-1\right )e^{-\epsilon t}}} +O(\epsilon ). \]

Raw LaTeX

Consider the scalar equation
\[
  \ddot{u}+\epsilon(u^2-1)\dot{u}+u=0,\qquad 0<\epsilon\ll1.
\]
Apply the method of multiple scales, with variables \(t\) and \(\eta=\epsilon t\) to show
that if \(u(0)=a>0\) and \(\dot{u}(0)=0\) then for \(t=O(\epsilon^{-1})\),
\[
  u=\frac{2\cos t}{\sqrt{1+\left(\frac{4}{a^2}-1\right)e^{-\epsilon t}}}
  +O(\epsilon).
\]

Topic Tags

Perturbation and Multiple Scales Methods