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We study the uniqueness of positive solutions of the boundary-value problem $$displaylines{ u''+a(t)u'+f(t,u)=0 ,quad tin (0,b)cr u(0)=0,u(b)=0,, }$$ where 0 less than $b$ less than $infty$, $ain C^1[0,infty)$ and $fin C^1([0,infty)imes [0, infty), [0, infty))$ satisfy suitable conditions. The proof of our main result is based on the shooting method and the Sturm comparison theorem.