I have this eigenvalue problem:(adsbygoogle = window.adsbygoogle || []).push({});

[itex] \frac{\mbox{d}^2y}{\mbox{d}x^2}+\left(1-\lambda\right)\frac{\mbox{d}y}{\mbox{d}x}-\lambda y = 0 \ , \ x\in[0,1], \ \lambda\in\mathbb{R}

[/itex]

[itex] y(0)=0 [/itex]

[itex] \frac{\mbox{d}y}{\mbox{d}x}(1)=0 [/itex]

Then, I have to show that there exists only one eigenvalue [itex] \lambda [/itex], and find this eigenvalue and write the corresponding eigenfunctions.

Thus far, I have solved the ODE's characteristic equation

[itex] r^2+(1-\lambda)r-\lambda=0 [/itex].

This gives me two solutions

[itex] r=-1 [/itex] and [itex] r=\lambda [/itex].

Thus the solution to the ODE is

[itex]

y=c_1\exp(-x)+c_2\exp(\lambda x) \ , \ x\in\mathbb{R} \ , \ c_1, \ c_2\in\mathbb{R}

[/itex].

Can I now conclude that because we only have real solutions to the characteristic equation, only one eigenvalue exists?

Secondly, I am not completely sure how to find the sought eigenvalue. I know how to find the eigenvalue when I have a solution that involves [itex] \sin [/itex] and [itex] \cos [/itex], but here I am not sure how to do it. Could anyone give me a hint?

Thirdly, when I have the eigenvalue there should not be any problems in writing the corresponding eigenfunctions, or is it?

I would appreciate any help. I am not looking for a solution to my homework problem, but any hints to the problem mentioned above are welcome.

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# Eigenvalue problem

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