Questions: Separation of Variables for Boundary Value Problems

Question 1

After substituting V(x,y,z) = X(x)Y(y)Z(z) into Laplace's equation and dividing by XYZ, you obtain (X''/X) + (Y''/Y) + (Z''/Z) = 0. Why must each term independently equal a constant?

• A. Because the physical boundary conditions require constant separation
• B. Because each term depends only on one variable, and a function of x alone cannot compensate for changes in a function of y and z alone — so both must be constant
• C. Because Laplace's equation only has constant solutions in bounded regions
• D. Because the eigenvalues of the system must be real

Answer: (B)

The argument is purely one of variable independence. If (X''/X) were not constant, varying x would change its value — but (Y''/Y) + (Z''/Z), depending only on y and z, cannot adjust to compensate. The only way a function of x alone can always sum with functions of y and z alone to give zero is if each term is individually constant. This reduction to constants is the mathematical core of the technique.

Question 2

Boundary conditions require V = 0 at x = 0 and x = a. Which functions X(x) satisfy both conditions simultaneously?

• A. X(x) = e^(kx) for any real k
• B. X(x) = cos(nπx/a) for positive integer n
• C. X(x) = sin(nπx/a) for positive integer n
• D. Any linear combination of exponentials

Answer: (C)

To vanish at both x = 0 and x = a, X must be zero at both endpoints. sin(nπx/a) equals zero at x = 0 (since sin(0) = 0) and at x = a (since sin(nπ) = 0 for all integers n). Cosines fail because cos(0) = 1 ≠ 0. Real exponentials are positive definite and cannot vanish at two points. This selection of allowed eigenfunctions is exactly the job boundary conditions do in the method.

Question 3

Boundary conditions in separation of variables serve primarily to constrain the coefficients (Aₙ) of the final superposition; the eigenfunctions themselves are determined by the PDE alone.

• T. True
• F. False

Answer: False

False. Boundary conditions do two jobs: they first determine which eigenfunctions are allowed (selecting sin over cos, ruling out exponentials) and then — on remaining surfaces — determine the coefficients Aₙ via Fourier analysis. The PDE alone would admit infinitely many solutions; boundary conditions filter which ones are physically realized. This dual role is the key conceptual point of the method.

Question 4

Once an eigenfunction X(x) = sin(nπx/a) is found for a particular n, the general solution to the boundary value problem is that single eigenfunction with an appropriate coefficient.

• T. True
• F. False

Answer: False

False. There are infinitely many eigenfunctions — one for each allowed positive integer n — and the general solution is a superposition of all of them: V = ΣAₙ sin(nπx/a)·f(y,z). The coefficients Aₙ are then determined by matching the remaining boundary conditions via Fourier analysis. A single mode is generally not flexible enough to satisfy an arbitrary prescribed boundary value.

Question 5

In separation of variables, why are only discrete values of the separation constant allowed (e.g., n = 1, 2, 3, ...) rather than a continuous range of values?

Model answer: Boundary conditions impose simultaneous constraints that only certain spatial frequencies can satisfy. For example, the condition V = 0 at both x = 0 and x = a forces sin(nπx/a) to vanish at both walls — which only works when the argument equals a multiple of π at x = a, selecting integers n = 1, 2, 3, .... A non-integer value of n would produce a function that satisfies the ODE but fails at least one boundary condition. The allowed discrete values are the eigenvalues; the corresponding functions are eigenfunctions.

The analogy is to standing waves on a string fixed at both ends: only wavelengths that fit exactly between the endpoints are allowed. Continuous values of the separation constant correspond to 'modes' that don't fit the boundaries. The fact that boundary conditions quantize the solution spectrum is the mathematical connection between classical boundary value problems and quantum mechanics — eigenvalue equations arise in both contexts for the same reason.