Mr. Integral
Math notes/Differential Equations/Series, Boundary Values and Checkpoint

Boundary-Value Problems and Eigenvalues — Cheat sheet

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Core rule

λn=(nπ/L)2,\lambda_n=(n\pi/L)^2, yn=sin⁡(nπx/L),n≥1.y_n=\sin(n\pi x/L),\quad n\ge1.

Watch for

In engineering, eigenfunctions describe spatial mode shapes; time evolution enters through an associated dynamic model. Orthogonality of distinct sine modes helps decompose a shape or forcing. Do not confuse the free amplitude of an eigenfunction with a unique boundary-value solution.