= Confined linear-pressure spherical magnetic equilibrium
{title2=$ (x^2-3)\sin x+3x\cos x=0$}
Set $\psi=\alpha\chi$, $p=p_0+P\chi$ and $\chi=F(r)\sin^2\theta$. A regular family is $F=A[\cos(\alpha r)-\sin(\alpha r)/(\alpha r)]+cr^2$, where $P=-\alpha^2c/\mu_0$. A zero <magnetic field> at radius a requires $F(a)=F'(a)=0$. For nonzero A this yields the displayed condition with $x=\alpha a$ and $c=-A[\cos x-\sin x/x]/a^2$. The boundary pressure is $p_0$, independent of angle. The trigonometric pair is tied by regularity and the radial differential equation; its coefficients are not independent.
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