# Finding an equation satisfied by the negative Eigen values

I have a system $$\frac{d^2y}{dx^2}+\lambda{y} =0$$ subject to $2y(0)+y'(0)=0$ and $y(1)=0$

Im trying to find the equation satisfied by negative eigenvalues. Heres what ive tried:

let $\lambda=-\alpha^2<0$

then, $y(x)=Acosh(\alpha{x})+Bsinh(\alpha{x})$ and $y'(x)=A{\alpha}sinh(\alpha{x})+B{\alpha}coshh(\alpha{x})$

now using the constraints I get a transcendental equation

$tanh(\alpha)=\dfrac{\alpha}2$

And this is where I'm stuck. I've plotted a graph with $tanh(\alpha)$ and $\dfrac{\alpha}2$ and I can see that there are two intersections, one at zero and another positive one, but I don't really know if that helps me. Am I right in saying $\lambda_{\alpha}=-{\alpha^2}$ and eigen value? if so how do I find the corresponding eigenfunction?

• I think you can't really say much more. – mookid May 2 '15 at 11:16
• is there a corresponding eigenfunction? – sean May 2 '15 at 12:11
• of course there is – mookid May 2 '15 at 12:19
• could you show me how to find it? – sean May 2 '15 at 13:07
• this is just the general solution where you include the constrains, – mookid May 2 '15 at 13:10