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We consider the Cauchy problem for the Emden–Fowler equation $$y^{\prime {}\prime }-x^ay^{\sigma }=0 $$ with parameters $$a\in \mathbb {R} $$ and $$\sigma Click to show full abstract
We consider the Cauchy problem for the Emden–Fowler equation $$y^{\prime {}\prime }-x^ay^{\sigma }=0 $$ with parameters $$a\in \mathbb {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 \in (-1,0)$$ .
Journal Title: Differential Equations
Year Published: 2021
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