= Stationary bending warp of a Kerr disk
{title2=$W=W_\infty\cos(x_{\mathrm{in}}-x)/\cos x_{\mathrm{in}},\quad x=4\sqrt{24a}\,r^{-5/4}/(5\epsilon)$}
For $a>0$, $\Sigma\propto R^{-3/4}$, constant <disk aspect ratio> $\epsilon=H/R$, and $r=Rc^2/(GM)$, the inviscid steady <disk warp> equation reduces to $W_{xx}+W=0$ in the displayed coordinate. The vanishing inner torque gives $W_x(x_{\mathrm{in}})=0$; the outer tilt gives $W(0)=W_\infty$. The displayed solution holds when $\cos x_{\mathrm{in}}\ne0$. If $x_{\mathrm{in}}=\pi$, it is $W=W_\infty\cos x$, with an aligned node at $r=2^{4/5}r_{\mathrm{in}}$ and opposite signed inner tilt. For negative spin the corresponding equation is hyperbolic, giving $W=W_\infty\cosh(x_{\mathrm{in}}-x)/\cosh x_{\mathrm{in}}$ with $x=4\sqrt{24|a|}\,r^{-5/4}/(5\epsilon)$.
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