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?