(a) Determine the type of the critical point at (0, 0) when > 0, =...

(a) Determine the type of the critical point at (0, 0) when μ > 0, μ = 0, μ < 0.

(b) Rayleigh equation. Show that the Rayleigh equation

0) " class="fr-fic fr-dib">also describes self-sustained oscillations and that by differentiating it and setting y = Y' one obtains the van der Pol equation.

(c) The Duffing equation is

where usually |β| is small, thus characterizing a small deviation of the restoring force from linearity. β > 0 and β < 0 are called the cases of a hard spring and a soft spring, respectively. Find the equation of the trajectories in the phase plane.For β > 0 all these curves are closed.)

Transcribed Image Text:

Y" – µ(1 – y'2)r' + Y = 0 (µ > 0) y" + way + By³ = 0 2. .3