Eigenvalues λₙ are determined via Sturm-Liouville analysis for the differential operator Ly. Corresponding normalized eigenfunctions ŷₙ are obtained by solving the Sturm-Liouville problem, ensuring orthogonality with respect to the weight function p(x) = e^x.
The given differential operator L is defined as Ly = (d/dx)(e^x(dy/dx)) - 4exy. To find the eigenvalues λₙ and corresponding eigenfunctions yₙ, we consider the eigenvalue problem Lyₙ = λₙe^xyₙ over 0 < x < 1, with boundary conditions y(0) = 0 and dy/dx + 1/2 * y = 0 at x = 1.
Assuming yₙ = e^(x/2)uₙ(x), we substitute this into the eigenvalue problem, leading to (d/dx)(p(x)duₙ/dx) + (λₙ - 1/4)e^xuₙ = 0, where p(x) = e^x.
The Sturm-Liouville form shows that eigenfunctions uₙ are orthogonal with respect to the weight function p(x) = e^x. Solving the Sturm-Liouville problem gives eigenvalues λₙ and unnormalized eigenfunctions uₙ(x). The orthogonality condition leads to the normalization factor. Finally, normalized eigenfunctions ŷₙ(x) are obtained.
In conclusion, eigenvalues are determined through Sturm-Liouville analysis, and corresponding normalized eigenfunctions ŷₙ are obtained, ensuring orthogonality with respect to the weight function p(x) = e^x.