We prove that the law of the minimum $m:=\min_{t\in[0,1]} ξ(t)$ of the solution $ξ$ to a one-dimensional ODE with good nonlinearity has continuous density with respect to the Lebesgue measure. As a byproduct of the procedure, we show that the sets $ \{ x\in C([0,1]):\; \min x > r\}$ have finite perimeter with respect to the law $ν$ of the solution $ξ(\cdot)$ in $L^2(0,1)$.