ON THE SOLUTIONS OF SECOND AND THIRD ORDER DIFFERENTIAL EQUATIONS

Abstract

This chapter presents the solutions of second- and third-order differential equations. It describes the nature of solutions of the second-order linear differential equation (ry')' + py = 0 to highlight the nature of the nonhomogeneous second-order linear differential equation (ry')' + py = f and of the third-order equation (ry″)' + py' = qy. When p(x), f(x), and r(x) are positive, having continuous derivative, p(x)r(x) is increasing and f(x)r(x) is decreasing, then the zeros of a solution y1(x) of equation (ry')' + py = 0 and a solution ya(x) of equation (ry')' + py = f, defined by y1(0) = ya(0) = 0, y'a(0) = a, seperate each other; moreover, the zeros of ya(x) for different values of a seperate each other. The chapter also presents results concerning the nth eigenfunction for an equation.