Find possible values for a in this differential equation

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!
Attached Thumbnails

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 Recognitions: Homework Help Science Advisor Well, what kind of solution forms are going to make it easy to show y(0)=0 and y(6)=0?
 I don't know what the thought process is for figuring that out. :(

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Find possible values for a in this differential equation

 Quote by s3a I don't know what the thought process is for figuring that out. :(
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?
 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?

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 Quote by s3a 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?
Right, now you are catching on. What kinds of values should b have in sin(bx) to make your boundary values work?
 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 Thumbnails

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