A string fixed at both ends admits sine-shaped modes; an insulated rod admits a constant mode as well as cosine modes. The difference is not cosmetic. The boundary conditions change the eigenvalues, the nullspace and the long-time behavior of the associated PDE.
Start with −y″ = λy on [0,π]. Impose either y(0) = y(π) = 0 or y′(0) = y′(π) = 0. Solve the λ = 0 case separately before trying trigonometric formulas.
Predict
For zero Dirichlet endpoints, can a nonzero affine function solve the zero-eigenvalue problem? For zero Neumann endpoints, can a constant do so? Predict how this distinction will affect heat decay.
Experiment
Direct substitution gives the following first modes.
Boundary condition
Modes
Eigenvalues
Dirichlet
sin x, sin(2x), sin(3x)
1, 4, 9
Neumann
1, cos x, cos(2x)
0, 1, 4
Check the endpoint conditions and the second derivative for each row. Then calculate the integral of sin x sin(2x) over [0,π] to see orthogonality concretely.
Understand
A regular Sturm–Liouville equation is −(p y′)′ + qy = λwy on a finite interval, with p and w positive and sufficiently regular real coefficients. The weighted inner product is ∫w yz. Suitable separated homogeneous boundary conditions make the boundary form p(y z′ − y′z) vanish at the endpoints.
Multiply the equation for y by z and the equation for z by y, subtract and integrate. The interior derivative terms reduce to the boundary form. If λ and μ are distinct eigenvalues and the form vanishes, (λ − μ)∫w yz = 0, proving weighted orthogonality. Omitting the weight would generally give the wrong statement.
For Dirichlet data, multiply an eigenfunction equation by y and integrate by parts to obtain λ = [∫p(y′)² + qy²]/∫wy². With q nonnegative this is nonnegative, and strict positivity follows under the usual positive p and Dirichlet conditions. General Robin conditions contribute boundary terms; arbitrary Robin coefficients do not guarantee positive eigenvalues.
The regular self-adjoint spectral theorem supplies a discrete real spectrum and a complete weighted L² eigenbasis under its stated coefficient and boundary hypotheses. We have proved the orthogonality and energy mechanisms, not the entire compact-operator theorem establishing completeness. Later separation arguments explicitly rely on that theorem when the modes are not elementary trigonometric ones.
Apply
For −y″ on [0,L], derive the Dirichlet eigenvalues (nπ/L)² and explain why increasing L lowers every frequency scale. Then propose a boundary condition that admits a constant eigenfunction. Check whether its boundary form vanishes and whether its zero mode must be treated separately in a time-dependent PDE.
Continue
Heat evolution assigns a decaying time factor to each positive eigenvalue. The next lesson combines those factors with initial-data projections and checks precisely what happens at time zero.
Apply · explain · revise
Practice and reasoning
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For −X″=λX on (0,π), X(0)=X(π)=0, what is the third eigenvalue?
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
For −X″=λX on (0,π) with zero derivatives at both endpoints, what is the smallest eigenvalue?
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
Prove eigenfunctions with distinct eigenvalues of −(pX′)′+qX=λwX are w-orthogonal under self-adjoint separated boundary conditions.
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Derive the Dirichlet Rayleigh quotient when p>0,w>0 and q≥0.
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State the regular Sturm–Liouville completeness result without claiming it follows from orthogonality alone.
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Explain how Robin conditions alter boundary terms and why arbitrary signs can affect positivity.
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Reflect and continue
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