I am having difficulty using the Rayleigh quotient to find the principal eigenvalue of the following SL problem
$$ u''(x) + (\lambda - x^{2})u(x) = 0\\ u'(0) = u(1) = 0 $$
To my understanding I need to find $u(x)$ first and then I can use the solution to find the principal eigenvalue. However, solving the above SL problem has proven to be quite difficult as I need to use parabolic cylinder functions (according to wolfram). I was curious if someone could shed some light on what I am doing wrong.