Extending work of \textit{R. Bouabdelli} and \textit{B. Belaïdi} [``Results on shared values of entire functions and their homogeneous differential polynomials'', Int. J. Difference Equ. 8, No. 1, 3--14 (2013)], results are obtained on the growth of an entire function \(f\) which is a solution of a differential equation of the form \[P(f)-\alpha_1=(f^n-\alpha_2)e^{A},\] where \(A\) is a nonconstant polynomial, \(P(f)\) is a homogeneous differential polynomial of degree \(n=\gamma_p\), \(\alpha_1\) and \(\alpha_2\) are entire functions with order less than one, and \(f\not =\alpha_2\).