  Given a large, high-dimensional sample from a spiked population, the top
sample covariance eigenvalue is known to exhibit a phase transition. We show
that the largest eigenvalues have asymptotic distributions near the phase
transition in the rank-one spiked real Wishart setting and its general beta
analogue, proving a conjecture of Baik, Ben Arous and P\'ech\'e (2005). We also
treat shifted mean Gaussian orthogonal and beta ensembles. Such results are
entirely new in the real case; in the complex case we strengthen existing
results by providing optimal scaling assumptions. One obtains the known
limiting random Schr\"odinger operator on the half-line, but the boundary
condition now depends on the perturbation. We derive several characterizations
of the limit laws in which beta appears as a parameter, including a simple
linear boundary value problem. This PDE description recovers known explicit
formulas at beta=2,4, yielding in particular a new and simple proof of the
Painlev\'e representations for these Tracy-Widom distributions.
