Dear  Mohamed,

See for example  Solutions of complex linear differential equations in the unit disc  by  Saleh Omran,   

MATHEMATICAL COMMUNICATIONS 205

Math. Commun. 16(2011), 205-214.

Complex oscillation of differential polynomials generated by

analytic solutions of differential equations in the unit disc by 

Ting-Bin Cao, Lei-Min Li, Jin Tu and Hong-Yan Xu,

and

G.G. Gundersen, E.M. Steinbart, S. Wang,

The possible orders of solutions of linear differential equations with polynomial coeficients, Trans. Amer. Math. Soc. 350(3) (1998),1225-1247.

There are also  a major breakthrough concerning the initial Schoen Conjecture related to Complex differential equations (for hypebolic harmonic) in The Unit Disc.

Dear Mohamed,

The equation which describes  harmonic hyperbolic mappings (wrt.  hyperbolic  distance)  is an  important complex non-linerar equation.

In [1], Markovic shows that every quasisymmetric homeomorphism of the circle $\partial \mathbb{H}^2$ admits a harmonic hyperbolic   quasiconformal extension to the hyperbolic plane $\mathbb{H}^2$. This proves the Schoen Conjecture. The Proof is based on:

Theorem 1.1. For every $K > 1$ there exists a constant $C = C(K)$ with

the following properties. We let $g : \mathbb{H}^2 \rightarrow \mathbb{H}^2$ be any K-quasiconformal

homeomorphism and assume that there exists a harmonic quasiconformal

homeomorphism $f : \mathbb{H}^2 \rightarrow \mathbb{H}^2$ that agrees with g on the boundary $\partial \mathbb{H}^2$.

Then

$\sup_{z\in \mathbb{H}^2} d(h(x),f(x)) \leq C$,

where $d(\cdot, \cdot)$ denotes the hyperbolic distance.

See 

[1]  V. Markovic,  Harmonic maps and the Schoen Conjecture, March 7, 2015, http://www.its.caltech.edu/~markovic/M-schoen.pdf

best,MM