Homework Help: Boundary Value Problem and Eigenvalues

1. Oct 27, 2012

domesticbark

1. The problem statement, all variables and given/known data

$y'' +λy=0$

$y(1)+y'(1)=0$

Show that $y=Acos(αx)+Bsin(αx)$ satisfies the endpoint conditions if and only if B=0 and α is a positive root of the equation $tan(z)=1/z$. These roots $(a_{n})^{∞}_{1}$ are the abscissas of the points of intersection of the curves $y=tan(x)$ and $y=1/z$. Thus the eigen values and eigen functions of this problem are the numbers $(α^{2}_{n})^{∞}_{1}$ and the functions ${cos(α_{n}x)}^{∞}_{1}$ respectively.
2. Relevant equations

See above.

3. The attempt at a solution

So I already showed B=0, what I'm confused about is the α
part. I got $y(1)+y'(1)=0$ to be $Acos(α)-Asin(α)=0$ which means $tan(α)=1$. When I graph the functions of z and find their intersection, I don't get the same values of α as I would expect from what I just solve. I'm kind of just guessing at what the question is really even asking me, because I have not idea where a positive root could even come from. The picture provided makes it seem that α should just be the value of z where they intersect, but that does seem to be right either. Can anyone figure out where a positive root could even come from?

2. Oct 27, 2012

Alpha Floor

You forgot to multiply by "alpha" when deriving y'(1), that's where you're missing the alpha to get your tan(z)=1/z equation

3. Oct 27, 2012

domesticbark

Well I feel silly now. Thanks.