Below is a full set of notes to "Ordinary Differential Equations in the Complex Domain" by Haraoka, "Ordinary Differential Equations" by Coddington and Levinson, and "Rigid Local Systems" by Katz with emphasis on isomonodromy problems and the analysis of irregular singularities.
Full Notes (By Sections of the Textbook)
The main source of these notes is Yoshishige Haraoka's "Linear Differential Equations in the Complex Domain" with the main goal of being an introduction to Katz's theory of middle convolution and monodromy data of linear differential equations.
Chapter 2: Linear Scalar Differential Equations
It is a classical fact that nth order scalar, homogeneous, linear ordinary differential equations (this is a mouthfull so from now on when I say "ODE" this is the kind I am talking about) are equivalent to first order nxn matrix systems of ODEs. It's easy to see that you can write a linear nth order ODE as a first order nxn matrix system; just take z_1 = z, z_2 = z', z_3 = z'',..., z_n = z^(n-1) and the matrix system is simple. The more challenging direction is to show that a first order nxn matrix system is equvalent to an nth order ODE. This involves the theory of cyclic vectors in the general case and involves a nontrivial transformation of the solution vector which we discuss!
Chapter 3: Analysis at Regular Points
The most well-behaved points on the complex plane in the domain of the solution space of an nth order ODE are those at which the coefficients are holomorphic. This is clear, as one would hope, since they are points where the solution is "the most rigid". Holomorphicity demands extremely restrictive properties of the solutions and, in particular, after giving initial conditions, gives existence and uniqueness of solutions. This rigidity gives two important corollaries: 1) Solutions can be analytically continued anywhere along curves in the domain of holomorphy of coefficients and 2) The vector space of all solutions of an nth order ODE is an n-dimensional complex vector space.