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

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Problem 5.5[edit]

Part 1[edit]

Problem Statement[edit]

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

Solution[edit]


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[edit]

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

Solution[edit]




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[edit]

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

Solution[edit]

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;

 

R5 code.jpg

R5 plot.jpg