On the perturbation of vibrating spring

In this work we deal with the nonlinear o.d.e u"+ku = εu3 which represents a spring-mass system with no damping but perturbed by external force εu3. We want to know how the spring constant k and the perturbed term act on the equation. So we study this equation by the way:(I) u" + ku = 0 (II)u" = u3 (III) u" + ku = εu3During the period of calculating, we find that k, ε and energy constant E(0) play important roles in the properties of the solutions of the equation. Finally we give the relation about them