On Existence of Singular Solutions of $n$-th Order Differential Equations

Miroslav Bartusek

Address. Department of Mathematics, Faculty of Science, Masaryk University, Janackovo nam 2a, 662 95 Brno, Czech Republic

E-mail: bartusek@math.muni.cz

Abstract. In the paper sufficient conditions are given under which the equation $ y^{(n)} = f(t, y, \dots , y^{(n-2)}) g (y^{(n-1)})$ has a singular solution $y: [T, \tau) \to \mathbb R, \tau <\infty$ fulfilling $\lim_{t\to \tau_-} y^{(i)} (t) = c_i \in \mathbb R, i=0,1,2, \dots , n-2$ and $\lim_{t\to\tau_-}|y^{(n-1)} (t) | = \infty$.

AMSclassification. 34C11

Keywords. Singular solutions, black hole solutions.