On the number of independent solutions of algebraic differential equations

We prove a conjecture of Kumbhakar, Roy, and Srinivasan (2024) on the classification of order one differential equations, and a conjecture of Kumbhakar and Srinivasan (2025) on higher order equations.

Both conjectures involve bounds for the number of independent solutions of the equation and are shown to be results of recent work in differential Galois theory using model theoretic techniques.

In both cases, stronger versions of the conjectures hold when working over the field of constants (i.e., when the equation is autonomous).

We then use inverse Galois theory to show that the bounds in the conjectures are optimal when working over a differential field which is differentially finitely generated over its constant subfield.

We also show how recent results of Jaoui and Moosa (2024) on abelian reductions of differential equations can be used to recover some of the work of Kumbhakar and Srinivasan (2025).