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Counterexample

Is this true?

Every boundary value problem has a Green's function

No — it is false.

The counterexample

y′′+π2y=fy'' + \pi^{2}y = f on [0,1] with y(0) = y(1) = 0

Why it fails

λ=π2\lambda = \pi^{2} is an eigenvalue of the homogeneous problem, so the operator is not invertible and no Green's function exists.

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Execution slip

The idea was right. The computation was not.

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Go deeper· Explained in full in Sturm–Liouville problems and Green's functions· Asked in CSIR NET December 2025· From Ordinary Differential Equations › Sturm–Liouville problems and Green's functions

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