Sturm's Theorems for Differential Equations with Singularities Legendre's Equation

Literature contains no special theorems on differential equations with singularities. It contains no theorems of the Sturm’s type which would apply to canonic interval between singular points, either. And finally, there are no theorems on existence of the solutions of differential equation close to each singularity. Whether equation with singularity has oscillating solutions, primarily depends on the sign of the coefficient of the equation, monotony or non-monotony of the coefficient in the interval between two singularities, type of the singularity, as well as on the width of the interval between singularities. In this work we have tried and hopefully succeeded in determining conditions under which Legendre’s equation (x2 1)  y  xy  n2 y  0 , which is differential equation with singularity, has oscillating solutions. In this way we have formed some premises on zeros of oscillations.