hw3[1] - z n of the transcendental equation sin 2 z = z 2...

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MAE294B/SIO203B: Methods in Applied Mechanics Winter Quarter 2010 http://maecourses.ucsd.edu/mae294b Homework III Due January 28, 2010. Questions with a star have a numerical/plotting component. 1 Investigate the behavior of the system ˙ x = ( μ - x 2 ) ± 1 - 2 x + μ 2 ² . In particular discuss the values of μ at which the qualitative nature of the solution changes. 2 Kepler’s equation nt = u - ε sin u relates the eccentric anomaly u to the eccentricity ε and the mean anomaly nt . Find the first three terms in the expansion of u for small ε . 3* Find approximations to the complex solutions
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Unformatted text preview: z n of the transcendental equation sin 2 z = z 2 that are valid for large | z | . Plot them along with the numerically-obtained solutions. (These values are related to the Papkovich–Fadle eigenfunctions of the biharmonic operator.) 4 Find leading-order uniform asymptotic approximations to ( a ) ε y 00 + y +( tanh x ) y = , y ( ) = y ( 1 ) = 1 , ( ≤ x ≤ 1 ) , < ε ± 1 , ( b ) ε y 00-1 1 + x 2 y + y = , y ( ) = y ( 1 ) = 1 , ( ≤ x ≤ 1 ) , < ε ± 1 . 1...
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