hw1_09 - ) ( ; 1 ) ( = = dt dx x Use the ode45 function to...

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01/20/09 EP 471 – Homework #1 Each problem is equally weighted: (1) Consider the first order IVP: 1 ) 0 ( ; = + + = y xy y x dx dy We wish to obtain the solution y ( x ) over the interval 2 0 x . Using the Euler, Modified Euler and Second Order Runge-Kutta methods we discussed in the first in-class exercise, find solutions to this IVP using a constant step size h = 0.1. Compare your results to a solution using Matlab’s ode45 function. Post-process your results to plot the step size Matlab chooses versus increment number. What was the minimum and maximum step size Matlab chose over the interval? (2) A spring system has nonlinear resistance to motion proportional to the square of the velocity, and its motion is described by: 0 6 . 0 1 . 0 2 2 2 = + + x dt dx dt x d If the spring is released from rest at a unit distance above its equilibrium point, the appropriate initial conditions are:
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Unformatted text preview: ) ( ; 1 ) ( = = dt dx x Use the ode45 function to find and plot x ( t ). How long does it take for the springs motion to damp out such that the amplitude x never again rises above one percent of its initial value? (3) The motion of the linear, undamped, compound spring system shown below is described by the pair of simultaneous second order equations: ) ( ), ( 2 1 2 2 2 2 2 2 1 2 1 1 2 1 2 1 y y k dt y d m y y k y k dt y d m = = where y 1 and y 2 are the displacements of the two masses from their equilibrium positions. Given m 1 = 1, m 2 = 2, k 1 = 3 and k 2 = 2, and the initial conditions: ) ( 2 ) ( ) ( 1 ) ( ' 2 2 ' 1 1 = = = = y y y y use ode45 to find and plot y 1 vs t and y 2 vs t and also to plot y 1 vs y 2 . k 1 m 2 m 1 k 2 y 1 y 2...
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