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**Ackbeet** Saw this problem in Zill, 6th Edition (it's Problem 4.3.62, in case you're curious), and had enough fun solving it that I thought I'd post it here for your pleasure. It's not an overly challenging problem, but it is more subtle than it appears at first.

What conditions should be imposed on the constant coefficients $\displaystyle a, b,$ and $\displaystyle c$ in order to guarantee that all solutions of the second-order differential equation $\displaystyle ay''+by'+cy=0$ are bounded on the interval $\displaystyle [0,\infty)?$ (Assume the coefficients are all real.)