# Definition of Eigenvalue for ODE

by Colin Hicks   Last Updated August 14, 2019 15:20 PM

I am confused about the usage of the terms eigenfunctions and eigenvalues for ODE’s. I am used to thinking of eigenvalues for some linear operator $$D$$ as the solutions $$D y=\lambda y$$ for some eigenfunction $$y$$. But I’ve seen in places that say an eigenvalue $$\lambda$$ satisfies something such as $$y''+ \lambda y=0,$$ but wouldn’t the eigenvalue be $$- \lambda$$ in this case for linear operator $$\frac{d^2}{dx^2}$$ not $$\frac{d^2}{dx^2} + \lambda$$ with eigenvalue $$\lambda?$$ This seems like a minor nuance, but is the second usage simply an informality that simplifies the analysis of ODE’s or is there an alternative I am missing?

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