In this paper, we investigate the existence and form of meromorphic solutions for the nonlinear differential equation \(f^{n}f^{(k)}+P_{d}(z,f)=p(z)e^{\alpha (z)}\) , where n, k are positive integers, \(P_{d}(z,f)\) is a differential polynomial in f(z) of degree d with small functions of f(z) as its coefficients, p(z) is a nonzero small function of f(z) and \(\alpha (z)\) is a nonconstant polynomial. Using this result, we prove that if f(z) is a transcendental entire function, then \(f^{n}f^{(m)}+q_{d}(f)\) assumes every complex number a infinitely many times, except for a possible value, where n, m are positive integers and \(q_{d}(f)\) is a polynomial.
