On the Existence and Stability of Periodic Solutions of Airy’s Equation with Elastic Coefficients

Abstract

In this paper, the existence and stability of periodic solutions of a certain second order differential equation with elastic coefficient were investigated using power series method, eigenvalue approach and lyapunov direct method. Existence of analytical solution which is independent of time was achieved using the power series method. Eigenvalue approach and Lyapunov direct method were used to investigate the stability of the resulting solution. Periodic solution was obtained using the eigenvalues of the resulting matrix. The first stability method further examined stability of the equilibrium point by considering the intervals around the origin and it’s discriminate. The equilibrium points for the intervals and the discriminate were unstable because the real part of the characteristics root is zero. Unstable equilibrium point was also obtained for the second stability method using the energy function and time derivative around the equilibrium point. The two unstable results indicated that there were highly instability regions with a strictly positive elastic coefficient. The highly instability regions were confirmed by the presence of elastic coefficient which reduces oscillation with an increase in amplitude. Furthermore, numerical simulations for existence and stability of Airy’s equation at different values of the elastic coefficient were illustrated in order to demonstrate the behaviour of the solutions which extends some results in literature.