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Behavior of solutions of $f''+Af=0$ is discussed under the assumption that$A$ is analytic in $\mathbb{D}$ and$\sup_{z\in\mathbb{D}}(1-|z|^2)^2|A(z)|<\infty$, where $\mathbb{D}$ is the unitdisc of the complex plane. As a main result it is shown that such differentialequation may admit a non-trivial solution whose zero-sequence does not satisfythe Blaschke condition. This gives an answer to an open question in theliterature.  It is also proved that $\Lambda\subset\mathbb{D}$ is the zero-sequence of anon-trivial solution of $f''+Af=0$ where $|A(z)|^2(1-|z|^2)^3\, dm(z)$ is aCarleson measure if and only if $\Lambda$ is uniformly separated. As anapplication an old result, according to which there exists a non-normalfunction which is uniformly locally univalent, is improved.

Solutions of complex differential equation having zeros on pre-given sequences