  Eigenvalue problems for linear differential equations, such as
time-independent Schr\"odinger equations, can be generalized to eigenvalue
problems for nonlinear differential equations. In the nonlinear context a
separatrix plays the role of an eigenfunction and the initial conditions that
give rise to the separatrix play the role of eigenvalues. Previously studied
examples of nonlinear differential equations that possess discrete eigenvalue
spectra are the first-order equation $y'(x)=\cos[\pi xy(x)]$ and the first,
second, and fourth Painlev\'e transcendents. It is shown here that the
differential equations for the first and second Painlev\'e transcendents can be
generalized to large classes of nonlinear differential equations, all of which
have discrete eigenvalue spectra. The large-eigenvalue behavior is studied in
detail, both analytically and numerically, and remarkable new features, such as
hyperfine splitting of eigenvalues, are described quantitatively.
