For the boundary conditions given below, obtain a functional Λ(y) whose stationary values give the eigenvalues of the equation
(1+x)\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } +(2+x)\frac { dy }{ dx } +\lambda y=0, y(0)=0, {y}^{\prime}(2)=0,
Derive an approximation to the lowest eigenvalue {λ}_{0} using the trial function y(x) = x{e}^{−x/2}. For what value(s) of γ would
y(x)=x{e}^{-x/2}+\beta \sin {\gamma x}
be a suitable trial function for attempting to obtain an improved estimate of {λ}_{0}?