Boundary Values & Fourier Series · Superchallenge problem

Consider $y'' + \lambda y = 0$ on $[0, 1]$ with the coupling conditions $y(1) = 2y(0)$ and $y'(1) = \dfrac12y'(0)$. (These make the problem self-adjoint, so all eigenvalues are real.) Find the smallest eigenvalue, rounded to four decimal places.