Boundary value problems and eigenvalues
Instead of starting conditions, many physical problems specify conditions at two ends. These boundary value problems lead to eigenvalues, the key to vibrations and waves.
Initial versus boundary conditions
An initial value problem fixes \(y\) and \(y'\) at one point and usually has a unique solution. A boundary value problem fixes conditions at two points, such as \(y(0) = 0\) and \(y(L) = 0\) for a string fixed at both ends. It may have one solution, none, or infinitely many.
An example with no solution
\(y'' + y = 0\), \(y(0) = 0\), \(y(\pi) = 1\). The general solution \(c_1\cos x + c_2\sin x\) with \(y(0) = 0\) gives \(c_1 = 0\), but then \(y(\pi) = c_2\sin\pi = 0 \ne 1\). No solution exists. The same equation with \(y(\pi/2) = 1\) has the unique solution \(\sin x\).
The eigenvalue problem
Consider \(y'' + \lambda y = 0\), \(y(0) = 0\), \(y(L) = 0\). For most \(\lambda\), only \(y = 0\) works. But for \(\lambda_n = \left(\frac{n\pi}{L}\right)^2\), the functions \(y_n = \sin\frac{n\pi x}{L}\) are nonzero solutions. These special values are eigenvalues and the solutions are eigenfunctions.
\lambda_n = \left(\frac{n\pi}{L}\right)^2,\quad y_n = \sin\frac{n\pi x}{L}Physical meaning
For a guitar string of length \(L\) fixed at both ends, the eigenfunctions are the shapes of its standing waves, and the eigenvalues determine the allowed frequencies: a fundamental and its overtones at integer multiples. That is why stringed instruments produce harmonic series of pitches.
Buckling of a column
A slender column under compressive load \(P\) satisfies \(EIy'' + Py = 0\) with \(y = 0\) at both ends. Nonzero (buckled) shapes appear only when \(P\) reaches \(\frac{\pi^2EI}{L^2}\), Euler's critical load. Below it the column stays straight; at it, the column can suddenly bow.
P_{\text{crit}} = \frac{\pi^2 EI}{L^2}Orthogonality
Eigenfunctions of such problems are orthogonal: \(\int_0^L \sin\frac{m\pi x}{L}\sin\frac{n\pi x}{L}\,dx = 0\) for \(m \ne n\). This property lets us expand functions as sums of eigenfunctions, the Fourier series that solve partial differential equations later in the course.
Sturm-Liouville theory
Many boundary value problems fit the Sturm-Liouville form \((py')' + qy + \lambda wy = 0\). The theory guarantees real eigenvalues, infinitely many of them, and orthogonal eigenfunctions. It unifies the vibrating string, heat flow, and quantum mechanics.
Common mistakes
Assuming a boundary value problem always has a unique solution. Forgetting to check the cases \(\lambda < 0\), \(\lambda = 0\), and \(\lambda > 0\) separately. Including \(n = 0\), which gives the trivial solution.
Key takeaways
Boundary value problems specify conditions at two points and may have zero, one, or infinitely many solutions. Eigenvalue problems have nonzero solutions only for special \(\lambda\). Eigenfunctions describe standing waves and buckling modes, are orthogonal, and form the basis of Fourier series.
Flashcard
Conditions at two points
Check your understanding
For y'' + lambda y = 0 with y(0) = y(pi) = 0, which value of lambda is an eigenvalue?