University of Florida/Egm4313/s12.team11.imponenti/R3.8

Problem 3.8

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solved by Luca Imponenti

Kreyszig 2011 pg.84 problem 5

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Problem Statement

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Find a (real) general solution. State which rule you are using. Show each step of your work.

 

Homogeneous Solution

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To find the homogeneous solution,  , we must find the roots of the equation

 

 

 

We know the homogeneous solution for the case of a double root to be

 

 

Particular Solution

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We have the following excitation

 

From table 2.1, K 2011, pg. 82, we have

 

Since this does not correspond to our homogeneous solution we can use the Basic Rule (a), K 2011, pg. 81 to solve for the particular solution

 

where

 

 

and

 

 

 

Plugging these equations back into the differential equation

 

 

 

 

from the above equation it is obvious that   and  

therefore the particular solution to the differential equation is

 

General Solution

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The general solution will be the summation of the homogeneous and particular solutions

 

 

    

The coefficients   and   can be readily solved for given either initial conditions or boundary value conditions.

Kreyszig 2011 pg.84 problem 6

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Problem Statement

edit

Find a (real) general solution. State which rule you are using. Show each step of your work.

 

Homogeneous Solution

edit

To find the homogeneous solution,  , we must find the roots of the equation

 

  with  

 

 

We know the homogeneous solution for the case of a double root to be

 

 

Particular Solution

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We have the following excitation

 

From table 2.1, K 2011, pg. 82, we have

 

Since this corresponds to our homogeneous solution we must use the Modification Rule (b), K 2011, pg. 81 to solve for the particular solution

so  

differentiating

 

 

 

and

 

 

grouping cosine and sine terms we get

 

and

 

next we substitute the above equations into the ODE

 

 

 

 

 

after cancelling terms; we can equate cosine and sine coefficients to get two equations

 

 

so   and  

and the particular solution to the ODE is

 

General Solution

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The general solution will be the summation of the homogeneous and particular solutions

 

 

    

Egm4313.s12.team11.imponenti 04:28, 21 February 2012 (UTC)