Boundedness of solutions to fourth-order differential equation with oscillatory restoring and forcing terms.

*(English)*Zbl 1141.34320Summary: This paper concerns the fourth order differential equation

$${x}^{\text{'}\text{'}\text{'}\text{'}}+a{x}^{\text{'}\text{'}\text{'}}+f\left({x}^{\text{'}\text{'}}\right)+g\left({x}^{\text{'}}\right)+h\left(x\right)=p\left(t\right)\xb7$$

Using the Cauchy formula for the particular solution of non-homogeneous linear differential equations with constant coefficients, we prove that the solution and its derivatives up to order three are bounded.

##### MSC:

34C11 | Qualitative theory of solutions of ODE: growth, boundedness |