Repeated Roots — Question 7

Question 7 – Solution

Strategy. Center the characteristic equation at its repeated root and compare the resulting real, repeated and complex formulas without conflating their global zero patterns.

Step 1: Solve each root case. The characteristic polynomial is $(r+1)^2-\mu$. If $\mu>0$, set $s=\sqrt\mu\in(0,1)$. The roots are $-1\pm s$ and the data give $$y_\mu=e^{-t}\frac{\sinh(st)}s.$$ At $\mu=0$, the root is $-1$ twice, and the solution is $\boxed{y_0=te^{-t}}$. If $\mu<0$, set $b=\sqrt{-\mu}>0$; the roots are $-1\pm ib$, giving $$y_\mu=e^{-t}\frac{\sin(bt)}b.$$ Each formula has value 0 and derivative 1 at zero and solves its stated equation.

Step 2: Recover the repeated-root limit from either side. For $t>0$, factor the first formula as $te^{-t}\sinh(st)/(st)$ and the third as $te^{-t}\sin(bt)/(bt)$. The supplied limits yield $y_\mu(t)\to te^{-t}$ from either side. At $t=0$, all values are zero directly. Thus the repeated-root solution is the common fixed-time limit, rather than a missing special value.

Step 3: Locate zeros before taking their limit. For $\mu\ge 0$, the displayed solution is strictly positive for every $t>0$, since $t>0$ and $\sinh(st)>0$. For $\mu<0$, its positive zeros are $$\boxed{t_n=\frac{n\pi}{\sqrt{-\mu}},\qquad n=1,2,\ldots}.$$ In particular, $t_1\to\infty$ as $\mu\uparrow 0$. The infinitely many oscillations move beyond every fixed finite observation window.

Step 4: Separate decay from zero counts. For $0<\mu<1$, both roots $-1\pm\sqrt\mu$ are negative; at zero, both repeated-root modes decay; for $\mu<0$, the exponential factor $e^{-t}$ bounds every fixed sine/cosine combination. Hence every solution decays throughout the stated parameter range.

On any fixed $[0,T]$, the ratios in Step 2 converge uniformly because their arguments lie in intervals shrinking to zero and both ratios extend continuously with value 1 there. Yet for every negative $\mu$ there are infinitely many later zeros, while $y_0$ has none for $t>0$. Convergence on fixed windows does not control zeros that escape to infinity.