Model Theory of Differential Fields and Ranks of Underdetermined Systems of Differential Equations

In this thesis we compute the Lascar rank for generic differential equations. First we examine the case of generic linear differential equations. In this case, we show that there is a definable bijection between the solution set of a generic underdetermined system of $k$ linear differential equations in $n \geq 2$ variables and $\mathbb{A}^{n-k}$. We explore how this result can be applied to non-generic linear differential equations. Next we consider the case of a generic non-linear differential equations. We show that the differential tangent space above a generic point is given by a generic linear differential equation. We compute the Lascar rank by utilizing the relationship between differential tangent spaces and the underlying variety combined with our result for generic linear varieties applied to the tangent space above a generic point.