Find possible values for a in this differential equation

s3a

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 Thumbnails

   

Dick

Well, what kind of solution forms are going to make it easy to show y(0)=0 and y(6)=0?

s3a

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

Dick

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?

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?

Dick

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

s3a

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.

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Dick

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?