Boundary Value Problem and Eigenvalues

[domesticbark]

Homework Statement

$y'' +λy=0$

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

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.

Homework Equations

See above.

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?

 

[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

 

[domesticbark]

Well I feel silly now. Thanks.