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Unformatted text preview: 6). (a) Introduce a new time scale = t and rewrite the ODE in this new time scale. (b) Introduce series expansions for x and 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 to eliminate secular terms. Hint: use the trigonometric identity cos 3 = 3 cos + cos 3 4 . (d) Show that the zeroth-order solution for the Dung oscillator is given by x ( t ) = a cos(1 + 3 8 a 2 ) t. 4. Apply the averaging method to the Dung oscillator x + x + x 3 = 0 to nd a 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 Poincar e-Lindstedt method....
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This note was uploaded on 01/04/2012 for the course CDS 140b taught by Professor List during the Fall '10 term at Caltech.
- Fall '10