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I'm really stuck with this problem. Can you help me please?

The equation $x=-tgx$ have infinite number of non negative solutions. Let's denote them by $x_1=0, x_2, x_3.$..etc.

Find eigenvalues and eigenvectors of:

$ \left\{\begin{matrix} y''+\lambda y=0\\ y(0)=0\\ y(1)+y'(1)=0 \end{matrix}\right.$

Thanks a lot!

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I agree with @JohnD: what exactly are you saying in the second paragraph? It seems completely adrift from the problem below it, for which I provided an outline of a solution. If I missed anything relevant because I did not understand that paragraph, please let me know. Also, why the "pde" tag? –  Ron Gordon Jan 21 '13 at 23:41
    
@rlgordonma 7 Thanks! I'm also don't understand the meaning of"The equation $x=−tgx$ have infinite number of non negative solutions. Let's denote them by $x1=0,x2,x3$...etc." the "pde" tag - by mistake, thank you! –  Lilly Jan 22 '13 at 7:14

2 Answers 2

up vote 2 down vote accepted

For the equation, you can immediately write the general solution as

$$y(x) = A \cos{\sqrt{\lambda} x} + B \sin{\sqrt{\lambda} x}$$

The condition $y(0)=0$ dictates that $A=0$. The condition $y(1) + y'(1) = 0$ dictates that the eigenvalues $\lambda_n$ satisfy

$$\sqrt{\lambda_n} + \tan{\sqrt{\lambda_n}} = 0$$

which, as @JohnD mentioned above, does not yield to simple, analytical solutions. Note, though, that we can say that, if we order the eigenvalues $\lambda_n$ in order of increasing values of $n$, then for large $n$, we should have that $\tan{\sqrt{\lambda_n}}$ becomes very large (in absolute value) and we can write

$$\lambda_n \sim \left [ (2 n+1) \frac{\pi}{2} \right ]^2 \; (n \rightarrow \infty) $$

The eigenfunctions are simply $\sin{\sqrt{\lambda_n} x}$.

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$x$ is missing in the eigenfunctions. –  Artem Jan 21 '13 at 23:47
    
@Artem: thanks for catching that. It seems I'd forget my head if it weren't attached. –  Ron Gordon Jan 21 '13 at 23:56

No idea what that second paragraph means, but standard ODE methods will yield the solution to the eigenvalue problem.

You won't be able to write down the eigenvalues explicitly, but you can implicitly. In practice, you can find as many of them as you need with a numerical root finding method of your choosing.

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What variables are being separated? To me, this looks like a straightforward, ordinary differential equation. –  Ron Gordon Jan 21 '13 at 23:11
1  
Sorry I meant to say "eigenfunction method commonly used in separation of variables problems" (and had just answered a sep of variables question right before this one). –  JohnD Jan 22 '13 at 0:00

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