Final answer:
The eigenvalues and eigenfunctions for the given boundary value problem can be found by solving the differential equation and applying the boundary conditions to derive conditions on λ that lead to non-trivial solutions. These solutions are expressed in terms of cosine functions with normalization constants.
Step-by-step explanation:
The question involves solving a boundary value problem (BVP) with the differential equation y'' + λ y = 0, subject to conditions y'(0) = 0 and y(1) = 0. Finding the eigenvalues and eigenfunctions in this context refers to determining the specific values of λ (lambda) for which non-trivial solutions exist, and corresponding functions (eigenfunctions) that satisfy the equation and boundary conditions. The solution to such a problem requires setting up the general solution for the differential equation and applying the boundary conditions to find specific values of λ that allow for non-trivial solutions (eigenfunctions).
The general solution to the differential equation is y(x) = A cos(√λ x) + B sin(√λ x). Applying the first boundary condition y'(0) = 0 will show that B must be zero due to the sine term. The second boundary condition y(1) = 0 leads to the determination of allowable values of λ, which are those values where cos(√λ) becomes zero, leading to the eigenfunctions y_n(x) = A_n cos(nπx) for n=1, 2, ... where A_n is a normalization constant determined by further conditions such as normalizability.