The exact meromorphic solutions of some nonlinear differential equations

Abstract

We find the exact forms of meromorphic solutions of the nonlinear differential equations

$${f^n} + q(z){{\rm{e}}^{Q(z)}}{f^{(k)}} = {p_1}{{\rm{e}}^{{\alpha _1}z}} + {p_2}{{\rm{e}}^{{\alpha _2}z}},\,\,\,\,n \ge 3,\,\,\,k \ge 1,$$

where q, Q are nonzero polynomials, Q ≡ Const., and p₁, p₂, α₁, α₂ are nonzero constants with α₁ ≠ α₂. Compared with previous results on the equation p(z)f³ + q(z)f″ = − sin α(z) with polynomial coefficients, our results show that the coefficient of the term f^(k) perturbed by multiplying an exponential function will affect the structure of its solutions.