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Find possible values for a in this differential equation 
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#1
Mar312, 04:58 PM

P: 580

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! 


#2
Mar312, 05:18 PM

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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?



#3
Mar412, 12:22 PM

P: 580

I don't know what the thought process is for figuring that out. :(



#4
Mar412, 12:37 PM

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Find possible values for a in this differential equation



#5
Mar412, 03:46 PM

P: 580

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? 


#6
Mar412, 03:58 PM

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#7
Mar412, 05:38 PM

P: 580

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. 


#8
Mar412, 05:47 PM

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