Linear Homogeneous Differential Equations — Question 7

Question 7 – Solution

Strategy. Evaluate the real modes at the sampling grid before interpreting the amount of information in the data.

Step 1: Read the visible coefficients. The polynomial is $(r^2+1)(r^2+4)$, so $$y=A\cos x+B\sin x+C\cos 2x+D\sin 2x.$$ At the sample times, $y(n\pi)=A(-1)^n+C$. Thus $A+C=3$ and $-A+C=1$, giving $A=1$ and $C=2$. These samples impose no restriction on $B,D$.

Step 2: Find the invisible subspace. Zero values at both even and odd sample indices force $A=C=0$, while the sine modes vanish at every $n\pi$. Therefore $$\boxed{\{y:y(n\pi)=0\ \forall n\ge 0\} =\operatorname{span}\{\sin x,\sin 2x\}.}$$ It has dimension two. Adding any member to one fitting signal leaves every recorded value unchanged, even though the functions differ between samples.

Step 3: Recover the missing oscillations. The derivative samples give $B+2D=1$ and $-B+2D=3$, hence $B=-1,D=1$. The unique signal is $$\boxed{y=\cos x-\sin x+2\cos 2x+\sin 2x.}$$ Its value samples are $(-1)^n+2$ and its derivative samples at $0,\pi$ are $1,3$, respectively. Every term solves the equation.

Step 4: Prove general identifiability. For arbitrary measurements $u=y(0)$, $v=y(\pi)$, $p=y'(0)$ and $q=y'(\pi)$, $$A=\frac{u-v}{2},\quad C=\frac{u+v}{2},\quad B=\frac{p-q}{2},\quad D=\frac{p+q}{4}.$$ These formulas exist and are unique for every real measurement vector. Value samples alone repeat only two linear constraints; their number does not restore the sine information. The figure displays the two independent invisible modes and their common sample zeros.