Find possible values for a in this differential equation

[s3a]

Homework Statement

The question is attached as Problem.jpg.

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

Homework Equations

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

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!

 

[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]

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?

 

[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]

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?

 

[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.

 

[Dick]

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.

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?