Find the Principal Eigenvalue of Sturm-Liouville Problem with Rayleigh Quotient

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Use the Rayleight quotient to find a good approximation for the principal eigenvalue of the Sturm-Liouville problem.
u'' + (\lambda - x^2)u = 0
0 < x < 0
u(0) = u'(1) = 0
Any help?
 
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Nobody has any idea?
 
http://gershwin.ens.fr/vdaniel/Doc-Locale/Cours-Mirrored/Methodes-Maths/white/
 
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neo143 said:
http://gershwin.ens.fr/vdaniel/Doc-Locale/Cours-Mirrored/Methodes-Maths/white/

Directions?


Back to the problem, I know that you multiply both sides by u, integrate by parts, and then solve for lambda, after that I'm stuck.
 
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Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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