hw11 - points(states if any 4 Consider the ode of Problem...

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ECE320 Homework 11 Spring 2006 Cornell University T.L.Fine We have our third prelim on Thursday, 27 April. Please hand in this assignment at the close of lecture on Tuesday, 25 April. Use only your assigned three-digit code and not your name. I will post solutions to this homework shortly after 1pm on Tuesday the 25th. Throughout, give reasons for your answers. 1. Classify the following ode s as to whether they are autonomous and what their orders and degrees are: ( a ) ¨ x + ω 2 x = 0 , harmonic oscillator; ( b ) ¨ x - tx = 0 , Airy ode; ( c ) t 2 ¨ x + t ˙ x + ( λ 2 t 2 - n 2 ) x = 0 , Bessel’s equation; ( d ) ˙ x - tx 2 = e - t ; ( e ) tan(¨ x ) - x = 0; ( f ) ¨ x ˙ xx = sin( t ); ( g ) ( ˙ x ) 2 - x = 0; ( h ) ¨ x = - x 2 + sin( ω ˙ x ) . 2. Represent each of the ode s of Problem 1 in the standard form ˙ y = F ( y ), if possible. 3. For each of the representations in Problem 2, determine the equilibrium
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Unformatted text preview: points (states), if any. 4. Consider the ode of Problem 1(h) with initial condition s (0) = ( x (0) , ˙ x (0)) = (-1 , 0) and ω = 2. (a) Numerically integrate this equation over the time domain D = [0 , 1] using a first-order Taylor’s series and time interval of .01 between steps. Submit a plot of x ( t ). (b) Repeat this task but use pplane7.m to perform the numerical integration process and submit a plot of x ( t ). (c) How similar are the results of (a) and (b)? 5. Repeat Problem 4 for ω = 20. How well do the results of your numerical integration compare with that of pplane7.m in this case? 1...
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