# how to show this DE has infinitely many eigenvalues

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• Nov 5th 2008, 06:12 PM
szpengchao
how to show this DE has infinitely many eigenvalues
$y''+\lambda y=0 \ \ \ \ \ \ y(0)=0, \ \ y(1)+y'(1)=0$

has infinitely many eigenvalues $\lambda_{1}<\lambda_{2}<...$

and indicate the behaviour of lambda as n goes to infinity.
• Nov 6th 2008, 02:16 AM
Opalg
You really ought not to need much of a hint in order to do this. You should recognise $y''+\lambda y=0$ as a simple harmonic motion equation (assuming that $\lambda>0$), with solution $y=A\cos\mu x + B\sin\mu x$, where $\mu = \pm\sqrt\lambda$. The initial conditions will give you a condition on $\mu$, of the form $\tan\mu=-\mu$. This is not an equation that you can solve explicitly, but by drawing a graph of the tan function, you can see that there is a doubly-infinite family of solutions, which for large values of $|\mu|$ will be close to odd multiples of $\pi/2$.