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Differential Equations: Please Help Solve!!!!_27

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Date Posted: 7/24/2008 10:37:02 AM  Status: Live
Differential Equations: Please Help Solve!!!!_27
Course Textbook Chapter Problem
Differential Equations N/A N/A N/A
Question Details:
Please write out all steps of solving this problem. It helps me better understand. Thank you.
 
Solve the Equation:
 
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Date Posted: 7/24/2008 11:22:08 AM  Status: Live
Asker's Rating: Lifesaver   
Response:
Okay first you have to find the characteristic equation of the homogenous part (i.e set the equation to 0).

y''->r2
y(0) ->r0 ->1
∴r2 + 9=0
r = ±3i
 
(the above is standard when the solution to the characteristic equation roots are purely imaginary as  e raised to 0 is one (not zero). )
Now there are two ways of solving this equation:
i) Variation of parameters
ii) Method of undetermined coefficients

I am going to choose the first one.

First find the wronskian: -3sin2(3t)-3cos2(3t)=-3

Next use the standard formula making y1 = sin3t and y2=cos3t  (since it was put in that order when finding the wronskian)




once you get v1 and v2
substitute in the formula:
y=v1*y1+v2*y2

The last step is long and tedious I know. But the second method much more error prone.


 













 
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