Spherical bessel differential function.

mccoy1
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I was looking at the above equation here:
http://mathworld.wolfram.com/SphericalBesselDifferentialEquation.html
Which has the following equation:
{(d ²/dx²)+(d/dx)+[x²-(n+1/2)²] }z =0.
In my opinion, this equation is of the order n+1/2 but the website and books claim it's of the order of a 1/2. How can that be? In maths books the order is what ever is squared and the solutions are of that order.
Thank you.
 
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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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