It is shown that if
Q(
z) is a non-constant polynomial, then all non-trivial solutions of
y''+(
ez+
Q(
z))
y=0
have zeros with infinite exponent of convergence. Similar methods are used to settle a problem of M. Ozawa: if
P(
z) is a non-constant polynomial, all non-trivial solutions of
y''+
e−zy'+
P(
z)
y=0
have infinite order.
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