On the transcendental solutions of a certain type of nonlinear differential equations

In this paper, we shall utilize Nevanlinna value distribution theory to study the solvability of transcendental meromorphic function f(z) that satisfies the differential equations of the form: f(2)(z) + b(z)(L(f))(2) = a(z), where L(f) denotes a linear differential polynomial in f, a (z) and b(z) are nonzero small functions of f(z). As an application, the method developed here can be used, for instance, to obtain all the entire solutions of the nonlinear differential equation 4 f(3) (z) + 3 f" (z) = - sin 3z.