University of Florida/Egm4313/s12.team11.gooding/R5

Problem 5.5

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Part 1

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Problem Statement

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Show that   and   are linearly independant using the Wronskian and the Gramain (integrate over 1 period)

Solution

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One period of  
Wronskian of f and g
 

Plugging in values for  
   
 
 

 They are linearly Independant using the Wronskian.

 
 
 
 
 
 
 

 They are linearly Independent using the Gramain.

Problem Statement

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Find 2 equations for the 2 unknowns M,N and solve for M,N.

Solution

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Plugging these values into the equation given ( ) yields;
 
Simplifying and the equating the coefficients relating sin and cos results in;
 
 
Solving for M and N results in;

   

Problem Statement

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Find the overall solution   that corresponds to the initial conditions  . Plot over three periods.

Solution

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From before, one period   so therefore, three periods is  
Using the roots given in the notes  , the homogenous solution becomes;
 
Using initial condtion  ;
 
 
with  
 
Solving for the constants;
 
 
Using the   found in the last part;