  We introduce a fully nonlinear PDE with a differential form $\Lambda$, which
unifies several important equations in K\"ahler geometry including
Monge-Amp\`ere equations, J-equations, inverse $\sigma_{k}$ equations, and the
deformed Hermitian Yang-Mills (dHYM) equation. We pose some natural positivity
conditions on $\Lambda$, and prove analytical and algebraic criterions for the
solvability of the equation. Our results generalize previous works of G.Chen,
J.Song, Datar-Pingali and others. As an application, we prove a conjecture of
Collins-Jacob-Yau for the dHYM equation with small global phase.
