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I have the following Sturm-Liouville problem: $$\frac{d^2 y}{dx^2}+\lambda x^2y=0,$$ where $y(0)=0$ and $y(1)=0$.

I have solved this using MAPLE and found the exact solution to be: $$y(x)=c_1\sqrt{x}J_\frac{1}{4}(\frac{1}{2}\sqrt{\lambda}x^2)+c_2\sqrt{x}J_\frac{1}{4}(\frac{1}{2}\sqrt{\lambda}x^2).$$ Where J is the Bessel function.

I am told to use the "BesselJZeros" command in MAPLE to find the smallest eigenvalue, any help much appreciated.

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The first term vanishes at the origin but the second, which should read $c_2\sqrt{x}J_{-1/4}(\sqrt{\lambda}x^2/2)$, has a nonzero limit there. So the first boundary condition requires $c_2=0$. Then the second boundary condition says $J_{-1/4}(\sqrt{\lambda}/2)=0$ so you'll need the first zero of $J_{1/4}$.

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