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Find possible values for a in this differential equation

  1. Mar 3, 2012 #1

    s3a

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    1. The problem statement, all variables and given/known data
    The question is attached as Problem.jpg.

    The answers for a are:
    a_1 = 6.52415567780804
    a_2 = 7.34662271123215
    a_3 = 8.71740110027234


    2. Relevant equations
    Characteristic equation and its interpretation based on what the roots are.


    3. The attempt at a solution
    My attempt is attached as MyWork.jpg. Basically, assuming that I am right so far, I do not know how to proceed.

    Any help would be greatly appreciated!
    Thanks in advance!
     

    Attached Files:

  2. jcsd
  3. Mar 3, 2012 #2

    Dick

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    Well, what kind of solution forms are going to make it easy to show y(0)=0 and y(6)=0?
     
  4. Mar 4, 2012 #3

    s3a

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    I don't know what the thought process is for figuring that out. :(
     
  5. Mar 4, 2012 #4

    Dick

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    Well, you get solutions that are exponentials and exponentials times trig functions. Which seems like the better choice to satisfy y(0)=0 and y(6)=0?
     
  6. Mar 4, 2012 #5

    s3a

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    Well, having thought much more about, I am still confused but not as much and I'm thinking that if 25 - 4a > 0, both constants must be 0 which is not what we want so 25 - 4a < 0 must hold and then if I recall correctly from class (which is "cheating") then, I believe I must choose the trigonometric equation.

    But, I was hoping you could tell me the ins and outs because, I learn best by reading solutions to things and then going "Aha!" and then forgetting and then coming back and getting another "Aha!" and then it makes intuitive sense and I never forget again.

    Edit: Oh wait! I think I do see why it's the trigonometric equation!

    So now k_3 = 0 and I have to do something with y = k_4 * e^(αx) * sin(bx), right? If so, what exactly must I do now?
     
  7. Mar 4, 2012 #6

    Dick

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    Right, now you are catching on. What kinds of values should b have in sin(bx) to make your boundary values work?
     
  8. Mar 4, 2012 #7

    s3a

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    I fail to see what finding b will do without finding α.

    Having said that, I found b = 6πn as can be seen in the attachment.
     

    Attached Files:

  9. Mar 4, 2012 #8

    Dick

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    Once you know a value of b, you should be able to work back to find the corresponding value of a. 6*pi*n will work, but it's nowhere near the smallest value that will. How are you solving sin(bx)=0? What values can bx have?
     
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