University of Florida/Egm4313/s12.team11.imponenti/R2.9

Report 2, Problem 9

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Problem Statement

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Find and plot the solution for the L2-ODE-CC corresponding to

 

with  

and initial conditions  ,  

In another figure, superimpose 3 figs.:(a)this fig. (b) the fig. in R2.6 p.5-6, and (c) the fig. in R2.1 p.3-7

Quadratic Equation

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  with  

 

 

Homogeneous Solution

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The solution to a L2-ODE-CC with two complex roots is given by

 

where  

 

Solving for A and B

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first initial condition  

 

 

 

second initial condition  

 

 

 

 

 

so the solution to our L2-ODE-CC is

                       

Solution to R2.6

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After solving for the constants   and   we have the following homogeneous equation

 

Characteristic Equation and Roots

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We have a real double root  

Homogeneous Solution

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We know the homogeneous solution to a L2-ODE-CC with a double real root to be

 

Assuming object starts from rest

 ,  

Plugging in   and applying our first initial condition

 

 

Taking the derivative and applying our second condition

 

 

 

 

Giving us the final solution

                  

Plots

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Solution to this Equation

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Superimposed Graph

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Our solution:   shown in blue

Equation for fig. in R2.1 p.3-7:   shown in red

Equation for fig. in R2.6 p.5-6:  shown in green

 

Egm4313.s12.team11.imponenti 03:38, 8 February 2012 (UTC)