Unformatted text preview: Â§ 6). (a) Introduce a new time scale Ï„ = Ï‰t and rewrite the ODE in this new time scale. (b) Introduce series expansions for x and Ï‰ and derive equations for x and x 1 . What are the initial conditions for x and x 1 ? (c) Solve the equation for x 1 and use the freedom in the series expansion for Ï‰ to eliminate secular terms. Hint: use the trigonometric identity cos 3 Ï„ = 3 cos Ï„ + cos 3 Ï„ 4 . (d) Show that the zerothorder solution for the Duï¬ƒng oscillator is given by x ( t ) = a cos(1 + Â± 3 8 a 2 ) t. 4. Apply the averaging method to the Duï¬ƒng oscillator Â¨ x + x + Â±x 3 = 0 to ï¬nd a ï¬rst approximation ( i.e. the x term in the expansion) to the true solution. First, derive the averaged equations. Then, integrate these equations for a general set of initial conditions and compare the solution with the approximation obtained in class using the PoincarÂ´ eLindstedt method....
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 Fall '10
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 Limit sets, limit cycle, phase space, Limitcycle, new time scale, Bendixson

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