In this paper, the authors consider meromorphic solutions of nonhomogeneous differential equation \(f^{n}(f^{\prime}+af)+P_{d}(z,f)=u(z){\rm{e}}^{v(z)},\) where n is a positive integer, a is a nonzero constant, Pd(z, f) is a differential polynomial in f(z) of degree d with rational functions as its coefficients and d ≤ n − 1, u(z) is a nonzero rational function, v(z) is a nonconstant polynomial with v′(z) ≠ (n + 1)a, v′(z) ≠ −na and \(v^{\prime}(z)\ne-{(n+1)^{2}\over{n}}a\) . They prove that if it admits a meromorphic solution f(z) with finitely many poles, then \(\matrix{{f(z) = s(z){{\rm{e}}^{{{v(z)} \over {n + 1}}}}} & {{\text{and}}} & {{P_d}(z,f) \equiv 0,}}\) where s(z) is a rational function and sn[(n + 1)s′ + sv′] + (n + 1)asn+1 = (n + 1)u. Using this result, they also prove that if f(z) is a transcendental entire function, then fn(f′ + af) + qm(f) assumes every complex number α infinitely many times, except for a possible value qm(0), where n, m are positive integers with n ≥ m + 1 and qm(f) is a polynomial in f(z) with degree m.
