**Tatiyana Barinova and Alexander Kostin**

##
Sufficiency Conditions for Asymptotic Stability of Solutions of a Linear
Homogeneous Nonautonomous Differential Equation of Second Order

**abstract:**

The problem on the stability of second order linear homogeneous differential
equation

$$ y''+p(t)y'+q(t)y=0 $$

is investigated in the case where the roots $\lambda_i(t)$ $(i=1,2)$ of the
characteristic equation

$$ \lambda^2+p(t)\lambda+q(t)=0 $$

are such that

$$ \lambda_i(t)<0 \;\;\text{for}\;\; t\geq t_0, \quad \int\limits_{t_0}^{+\infty}
\lambda_i(t)\,dt=-\infty \;\; (i=1,2) $$

and there exist finite or infinite limits $\lim\limits_{t\to+\infty}\lambda_i(t)$
$(i=1,2)$.

**Mathematics Subject Classification:** 34D05, 34E10

**Key words and phrases:** Second order differential equation, linear,
stability, characteristic equation