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(Requesting Peer Review; Please kindly email me at bjn@brocknash.com)
Use your **Protective Helmet** to steer the timeline.
The **Painlevé VI** equation is a second-order nonlinear ordinary differential equation with the "Painlevé property," meaning its solutions have no movable singularities other than poles. Its full form is given by:
$$\frac{d^2y}{dt^2} = \frac{1}{2}\left(\frac{1}{y} + \frac{1}{y-1} + \frac{1}{y-t}\right)\left(\frac{dy}{dt}\right)^2 - \left(\frac{1}{t} + \frac{1}{t-1} + \frac{1}{y-t}\right)\frac{dy}{dt} + \frac{y(y-1)(y-t)}{t^2(t-1)^2} \left[ \alpha + \frac{\beta t}{y^2} + \frac{\gamma(t-1)}{(y-1)^2} + \frac{\delta t(t-1)}{(y-t)^2} \right]$$
The code in this visualizer leverages the **perspective operator** to simplify this complex formula. It is defined as:
$$a\triangleleft b:=\frac{a}{a-b}$$
Using this operator, the ODE can be written in a much more compact form. The shorthands used are $$Y_1:=y\triangleleft 1,\; Y_t:=y\triangleleft t,\; T_1:=t\triangleleft 1$$. The ODE is then:
$$y''=\tfrac12\frac{1+Y_1+Y_t}{y}(y')^2 -\Big(\frac1t+\frac{T_1}{t}+\frac{Y_t}{y}\Big)y' + \frac{y^3T_1^2}{t^4Y_1Y_t}\Big[\alpha+\beta\frac{t}{y^2}+\gamma\frac{tY_1^2}{y^2T_1}+\delta\frac{t^2Y_t^2}{y^2T_1}\Big]$$
**Perspective Calculus:**
$$\partial_a(a\triangleleft b)=-\frac{b}{(a-b)^2},\quad \partial_b(a\triangleleft b)=\frac{a}{(a-b)^2},\quad \frac{d}{dt}(a\triangleleft b)=\frac{a\dot b-b\dot a}{(a-b)^2}$$ $$\int (a\triangleleft b)\,da=a+b\ln|a-b|+C,\quad \int (a\triangleleft b)\,db=-a\ln|a-b|+C$$
For more information, see the Wikipedia article on Painlevé transcendents.