# Hokkaido Mathematical Journal

## Hokkaido Mathematical Journal, 39 (2010) pp.127-138

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### Abstract

This paper is devoted to studying the growth and the oscillation of solutions of the second order non-homogeneous linear differential equation $$f''+A_{1} (z) e^{P (z)}f'+A_{0} (z) e^{Q (z)}f = F,$$ where $P (z)$, $Q (z)$ are nonconstant polynomials such that $\deg P=\deg Q=n$ and $A_{j} (z)$ $( \not\equiv 0 )$ $(j=0,1)$, $F\not\equiv 0$ are entire functions with $\rho ( A_{j} ) < n$ $( j=0,1 )$. We also investigate the relationship between small functions and differential polynomials $g_{f} (z)=d_{2}f''+d_{1}f'+d_{0}f$, where $d_{0} (z)$, $d_{1} (z)$, $d_{2} (z)$ are entire functions that are not all equal to zero with $\rho ( d_{j} ) < n$ $( j=0,1,2 )$ generated by solutions of the above equation.

MSC(Primary) 34M10(MSC2000), 30D35(MSC2000) linear differential equations; entire solutions; order of growth; exponent of convergence of zeros; exponent of convergence of distinct zeros;