Chapter 4.5 Problem 14.Self Sustained Oscillations a.) Van der Pol equation ( y\

Question

Chapter 4.5 Problem 14.Self Sustained Oscillations a.) Van der Pol equation ( y''-m(1 - y^2 )y' + y = 0 ). Determine the type of critical point at (0,0) when m>0, m=0, m<0. b.) Rayleigh Equation. Show that the Rayleigh equation Y'' - m(1 - 1/3*(Y')^2 )Y' + Y = 0 (m > 0) also describes self-sustained oscillations and that by differentiating it and setting y = Y' one obtains the van der Pol equation. c.) Duffing Equation. The Duffing equation is y'' + (w^2)y + By^3 = 0 where usually |B| is small thus characterizing a small deviation of the restoring force from linearity. B>0 and B<0 are called the cases of hard spring and soft spring, respectively. Find the equation of the trajectories in the phase plane. (Note that for B>0 all these curves are closed.)

Explanation / Answer

you can use the formula: y = x tan (theta) - ( (g * x^2) / (2 v^2 cos^2(theta) ) where: y= the y coordinate of the end of the projectile x= the x coordinate of the end of the projectile theta= angle of inclination (from the horizontal) of the initial velocity g= gravitational constant, 9.81 v= initial velocity for this formula, you will place the initial point of the trajectory at the coordinates (0,0)

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