Some solutions of a nonlinear differential equation of high order

Some exact monotonic, and approximate oscillatory, solutions of the nonlinear equation ${d^n}y/d{x^n} = K{\left | y \right |^r} \operatorname {sgn} y$, $0 \le r$, are derived. The coefficient $K$ may be positive or negative, $r$ may be non-integral and $n$ is any positive integer. For the case $r = 0$, exact solutions in closed form are obtained. The conditions under which the approximate solutions will be highly accurate are discussed. Every component of the general solution of the linear equation ${d^n}y/d{x^n} = Ky$ is shown to be analogous to a corresponding solution of the given nonlinear equation.