Existence and Uniqueness of Solutions of Some Cauchy Problems for the Emden-Fowler Equation

We consider the Cauchy problem for the Emden-Fowler equation y '' x(a)y(sigma) = 0 with parameters alpha is an element of R and sigma < 0 for which the initial value of the solution belongs to one of the positive coordinate axes. Necessary and sufficient conditions on the parameters of the equation for the Cauchy problem with initial value on the positive ordinate axis to have a solution are obtained. (In this case, an improper initial value of the derivative of the solution is allowed.) The problem with initial value on the positive abscissa axis is shown to have a unique solution for sigma is an element of (-1, 0).