Second-Order Linear Equations · Superchallenge problem

A solution $y$ of $y'' + by' + cy = 0$, where $b$ and $c$ are real constants, is sampled once per second: $y(0) = 1$, $y(1) = 1$, $y(2) = 0$, $y(3) = -2$. Find the smallest possible value of $c$.