Elliptic and multiple-valued solutions of some higher order ordinary differential equations

Abstract

In the present paper, by the complex method, the meromorphic solutions of the higher order ordinary differential equation $ w^{(5)}+aw^{''}+bw^2-cw+d = 0 $ are investigated, where $ a, b, c, d $ are constant complex numbers, and $ b \neq0 $. Furthermore, by Theorem 1.1, we built elliptic and multiple-valued solutions for the higher order ordinary differential equations $ u^{(6)}-u^{(5)}+u'^2-2u'u+u^2+2u'-2u+1 = 0 $ and $ u^{(6)}-u^{(5)}+au^{'''}-au''+bu'^2-2bu'u+bu^2-cu'+cu+d = 0 $. At the end, we give some new meromorphic solutions for two higher-order KdV-like equations.

Keywords

• the complex method
• meromorphic solution
• elliptic function
• multiple-valued solution
• kdv-like equation