Solutions to Systems — Question 9

Question 9 – Solution

Strategy. Treat the unknown initial velocity as a parameter and ask whether the observations actually determine it.

Step 1: Parametrize all compatible initial states. Since $x''=-x$ and $x(0)=1$, the complete family is $$\boxed{x=\cos t+c\sin t,\qquad y=-\sin t+c\cos t.}$$ Direct differentiation verifies both rows. The arbitrary constant is $c=y(0)=x'(0)$, the missing initial component.

Step 2: Test the observation equation. The second measurement requires $\cos T+c\sin T=b$. If $\sin T\ne 0$, there is exactly one solution, with $\boxed{c=(b-\cos T)/\sin T}$. If $T=k\pi$ for a positive integer $k$, the measurement is independent of $c$: there are infinitely many solutions when $b=(-1)^k$, and none when $b\ne(-1)^k$. These cases exhaust all positive times.

Step 3: Evaluate the two specified experiments. At $(\pi,-1)$ every real $c$ is allowed, giving the full family in Step 1. The figure shows three of these indistinguishable endpoint measurements. At $(\pi/2,2)$, $c=2$, so the unique solution is $$\boxed{X=(\cos t+2\sin t,\,-\sin t+2\cos t)^T.}$$

Step 4: Quantify sensitivity and interpret it. Replacing $b$ by $b+\delta$ changes $c$ by exactly $\boxed{\Delta c=\delta/\sin T}$. Thus the absolute amplification factor is $1/|\sin T|$, which is unbounded as $T\to\pi$ through nonexceptional times. A fixed nonzero measurement error can then create a large recovered initial-velocity error. At $T=\pi$, that component is unobservable from these two measurements. Each fully specified initial vector still has a unique solution; the difficulty lies in recovering it from partial observations.