  We prove the Existential Closedness conjecture for the differential equation
of the $j$-function and its derivatives. It states that in a differentially
closed field certain equations involving the differential equation of the
$j$-function have solutions. Its consequences include a complete axiomatisation
of $j$-reducts of differentially closed fields, a dichotomy result for strongly
minimal sets in those reducts, and a functional analogue of the Modular
Zilber-Pink with Derivatives conjecture.
