= Painlevé II equation
{c}
{title2=$y''=2y^3+xy+\alpha$}
The <Painlevé II equation> is the nonlinear second-order <ordinary differential equation>
$$
y''=2y^3+xy+\alpha,
$$
where $\alpha$ is a constant <parameter>. This is the standard normalization of the second of the six <Painlevé equations>.<NIST Digital Library of Mathematical Functions>[equation32.2.2]{ref} A local solution is specified by $y(x_0)$ and $y'(x_0)$; as a <complex ordinary differential equation>, its analytic continuation exhibits the special singularity structure of the <Painlevé property>.
Special <parameters> can give elementary solutions: $y=0$ solves the equation at $\alpha=0$, while $y=-1/x$ solves it at $\alpha=1$ away from zero. For the latter, both sides equal $-2/x^3$. These are exceptional rational members of a generally non-elementary family of <Painlevé transcendents>.<NIST Digital Library of Mathematical Functions>[section32.8(ii), equation32.8.1 and seed solution at alpha0]{ref} <Similarity reductions> of some <integrable systems> lead to this equation after a change of scales.
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