Conditions for non-oscillation of singular linear differential equations of second order

Conditions are found in the fulfillment of which each non-trivial solution of the equation uPrime+ $u''+\beta(t)u'+\alpha(t)u=0$, where $\beta(t)\in L(a,b)$ and $(t-a)(t-b)\alpha(t)\in L(a,b)$ has not more than one zero on the interval $a\le t\le b$.