An extension of the general Struble's method for solving an \(n\)th order nonlinear differential equation when the corresponding unperturbed equation has some repeated eigenvalues
The authors show that the extended general Struble's technique is more suitable than the unified Krylov-Bogoliubov-Mitropolskii (KBM) method in solving problems that occur in critical conditions. First, the method is illustrated by the fourth-order nonlinear differential equation of the form \[ x^{(4)} + c_1 {x}''' + c_2 {x}'' + c_3 {x}' + c_4 x = - \varepsilon x^3, \] whose unperturbed equation has repeated complex eigenvalues. Second, a steady-state solution is also determined for the following fourth order nonlinear and non-autonomous equation of the form \[ x^{(4)} + c_1 {x}''' + c_2 {x}'' + c_3 {x}' + c_4 x = - \varepsilon x^3 + \varepsilon E\cos vt. \] Further, a critical condition is investigated when two real eigenvalues of the unperturbed equation are non-positive and equal. Some numerical examples are also given.
\(n\)th order nonlinear differential equation
scientific article; zbMATH DE number 5552686
scientific article