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We investigate an asymptotic behaviour of damped non-oscillatory solutions of the initial value problem with a time singularity ${\left(p\left(t\right)u^{\text{'}}\left(t\right)\right)}^{\text{'}}+p\left(t\right)f\left(u\left(t\right)\right)=0$, $u\left(0\right)=u_0$, $u^{\text{'}}\left(0\right)=0$ on the unbounded domain $[0,\infty )$. Function $f$ is locally Lipschitz continuous on $ℝ$ and has at least three zeros $L_0<0$, $0$ and $L>0$. The initial value $u_0\in (L_0,L)\setminus \left\{0\right\}$. Function $p$ is continuous on $[0,\infty ),$ has a positive continuous derivative on $(0,\infty )$ and $p\left(0\right)=0$. Asymptotic formulas for damped non-oscillatory solutions and their first derivatives are derived under some additional assumptions. Further, we provide...