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So I'm learning about eigenfunctions and eigenvalues and there appear to be 2 main forms

1) $L[y]=\lambda y$: This is intuitive to me as a direct extension of what I learned in Linear Algebra.

2) $L[y]+\lambda y=0$: This is the way that my textbook and several online resources teach it.

Form 1 makes perfect sense to me. You are given some $L[y]$ and are asked to find eigenfunctions/values. You make the eqn $L[y]-\lambda y=0$ and solve, using known methods and boundary conditions etc...

Form 2 confuses me. An example we worked with a lot was $y''+\lambda y=0$ with varying boundary conditions. Obviously, you can still find functions and values using similar methods, but you aren't finding eigen-stuff of $L[y]=y''$, because then the equation would be $y''-\lambda y=0$ (note the minus sign). What do these eigenfunctions/values mean? Why would you ever use Form 2 (ie just using $y''-\lambda y=0$ would be fine) and what linear operator are you finding eigen-stuff for when you solve Form 2?

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    $\begingroup$ Well, I'd guess it's not really that important which form you use - the latter simply defines eigenvalues to be the negatives of what the first defines. $\endgroup$ – George K May 12 '18 at 11:31
  • $\begingroup$ I agree with George. The second form is motivated by the need of writing equations in the form $\text{something}=0$. Sometimes it is better to write them in that way, to make it clear that they are homogeneous equations, or that there are no source terms, etc... $\endgroup$ – Giuseppe Negro May 12 '18 at 11:41

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