Exact number of local extreme points of curvature function of solutions of second-order linear differential equations

We study local properties of the curvature κ y ( x ) of every nontrivial solution y = y ( x ) of the second-order linear differential equation ( P ): ( p ( x ) y ′)′+ q ( x ) y =0, x ∈( a , b )= I , where p ( x ) and q ( x ) are smooth enough functions. It especially includes the Euler, Bessel and other important types of second-order linear differential equations. Some sufficient conditions on the coefficients p ( x ) and q ( x ) are given such that the curvature κ y ( x ) of every nontrivial solution y of ( P ) has exactly one extreme point between each two its consecutive simple zeros. The problem of three local extreme points of κ y ( x ) is also considered but only as an open problem. It seems it is the first paper dealing with this kind of problems. Finally in Appendix , we pay attention to an application of the main results to a study of non-regular points (the cusps) of the ε -parallels of graph Γ ( y ) of y (the offset curves of Γ ( y )).