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Problem 1. (25 points) Consider the eigenvalue problem (EVP) that arises in the analysis of reaction-diffusion systems. The mass balance equation of the species of interest in one spatial dimension X is described by the following equation at steady state: d2 Lc(x) = -72c(x), 4= dx2 - Da, 0 0 is the Damkohler number, a dimensionless quantity that measures the relative importance of reaction rate over diffusion rate. The model is completed by a Dirichlet boundary condition at x = 0, i.e., c(0) = 0, and a Neumann boundary condition at x = 1, i.e., c'(1) = 0. (a) Prove that the linear operator _ is self-adjoint by transforming it into an equivalent Sturm-Liouville operator. (5 points) (b) Examine whether the EVP (i.e., the ODE together with the homogeneous BCs) is self- adjoint. (5 points) (c) Solve the EVP, i.e., calculate the eigenvalues, 1, and the corresponding normalized eigenfunctions for 0