Quasiconformal surgery and linear differential equations

We describe a new method of constructing transcendental entire functions $A$ such that the differential equation $w"+Aw=0$ has two linearly independent solutions with relatively few zeros. In particular, we solve a problem of Bank and Laine by showing that there exist entire functions $A$ of any prescribed order greater than $1/2$ such that the differential equation has two linearly independent solutions whose zeros have finite exponent of convergence. We show that partial results by Bank, Laine, Langley, Rossi and Shen related to this problem are in fact best possible. We also improve a result of Toda and show that the resulting estimate is best possible. Our method is based on gluing solutions of the Schwarzian differential equation $S(F)=2A$ for infinitely many coefficients $A$.