Repeated Roots

A repeated root is simply what you get when the characteristic equation's two roots land on the same number, like $(r - 2)^2 = 0$ giving $2$ twice. It's the knife-edge between swinging and not swinging, which is exactly where engineers like to tune things such as car suspensions. All we're really saying here is that you need one more solution, and it's the same exponential multiplied by $t$. In $y = c_1e^{rt} + c_2\,t\,e^{rt}$, the pieces are $e^{rt}$ (the one exponential the root gives you), $te^{rt}$ (that same curve, stretched by a factor that grows over time) and $c_1$, $c_2$ (how much of each). That little extra $t$ is the only new thing in this lesson.

So far each root has given us one solution, and two roots gave us the two we need. A repeated root happens when $b^2 = 4ac$: the quadratic formula's square root is zero, and both roots land on the same number $r$. That leaves us one solution short. Writing $c_1e^{rt} + c_2e^{rt}$ doesn't help, since that's just $(c_1 + c_2)e^{rt}$, one constant in disguise.

So what can we do now? The fix is $te^{rt}$. You can check it directly: with a double root, $ar^2 + br + c = 0$ and also $2ar + b = 0$ (that's what makes the root double), and those two facts make every term cancel when you plug $te^{rt}$ in. The next lesson shows where it comes from.

Where it sits. A repeated root is the boundary between oscillating behavior (complex roots) and non-oscillating behavior (distinct real roots). For a spring, that boundary is called critical damping.

With $r < 0$, $te^{rt}$ still decays, because exponentials beat polynomials. But it can rise first, so the solution may overshoot once before settling. It never oscillates.

A common mistake. When you apply $y'(0)$, it's tempting to think the $c_2te^{rt}$ term contributes nothing, because it's zero at $t = 0$. Its derivative isn't: the product rule gives $c_2(1 + rt)e^{rt}$, which is $c_2$ at $t = 0$. So $y'(0) = rc_1 + c_2$.

Car suspensions and instrument needles are tuned near critical damping: the fastest return to rest without oscillating.