  Hrushovski's generalization and application of [Jouanolou, "Hypersurfaces
solutions d'une \'equation de Pfaff analytique", Mathematische Annalen, 232
(3):239--245, 1978] is here refined and extended to the partial differential
setting with possibly nonconstant coefficient fields. In particular, it is
shown that if $X$ is a differential-algebraic variety over a partial
differential field F that is finitely generated over its constant field F_0,
then there exists a dominant differential-rational map from X to the constant
points of an algebraic variety V over F_0, such that all but finitely many
codimension one subvarieties of X over F arise as pull-backs of algebraic
subvarieties of V over F_0. As an application, it is shown that the algebraic
solutions to a first order algebraic differential equation over C(t) are of
bounded height, answering a question of Eremenko. Two expected model-theoretic
applications to DCF_{0,m} are also given: 1) Lascar rank and Morley rank agree
in dimension two, and 2) dimension one strongly minimal sets orthogonal to the
constants are \aleph_0-categorical. A detailed exposition of Hrushovski's
original (unpublished) theorem is included, influenced by [Ghys, "\`A propos
d'un th\'eor\`eme de J.-P. Jouanolou concernant les feuilles ferm\'ees des
feuilletages holomorphes", Rend. Circ. Mat. Palermo (2)}, 49(1):175--180, 2000.
