SL diff eqn

\dfrac{d^2 y}{dx^2} + \lambda y = 0 , \ \ x \in [0, \pi] [br]\\ 1. y(0) = 0 [br]\\ 2. y'(\pi)+y(\pi)=0

Working

I'm asked to show that there exists an infinite sequence of positive eigenvalues (this is fine, each value of

\alpha

is reflected in y=-x on the graph and presto, we have countably infinite evalues in

\mathbb{Z}_{\geq0}

).

Then, I'm asked to show that the smallest eigenvalue

\lambda_1

is such that

\dfrac{1}{4} < \lambda_1 < 1

, whereas I've simply found it to be

\dfrac{1}{2} < \lambda_1 < 1

. I'm not sure what I've done wrong here.

Secondly, as n becomes large,

\lambda_n \rightarrow (n-\frac{1}{2})^2

... I'd have thought this was simply because each value of

\alpha_n

essentially "tends to" (for lack of a better phrase)

n - \frac{1}{2}

, and since

\lambda_n = \alpha_n^2

, we have

\lambda_{n} \approx {(n - \frac{1}{2})}^2

...?

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