Real & Distinct Roots — Question 5

Question 5 – Solution

Strategy. Recover the two exponential multipliers from a sample recurrence, then convert those multipliers back into differential-equation roots.

Step 1: Derive the recurrence. Each $z_j$ satisfies $z_j^2-(z_1+z_2)z_j+z_1z_2=0$. Multiplying by its mode coefficient and $z_j^n$, then summing, gives $$\boxed{S_{n+2}=sS_{n+1}-kS_n},\qquad s=z_1+z_2,\quad k=z_1z_2.$$

Step 2: Solve the sample equations exactly. For $n=0,1$, $$s-3k=3/8,\qquad (3/8)s-k=5/32.$$ Substitution of $s=3/8+3k$ into the second equation gives $k/8=1/64$. Hence $$\boxed{k=1/8,\qquad s=3/4}.$$ The nonzero coefficient $1/8$ in this elimination proves uniqueness. Equivalently, $S_0S_2-S_1^2=1/8\ne 0$.

Step 3: Recover and verify the continuous model. The multipliers solve $z^2-(3/4)z+1/8=(z-1/2)(z-1/4)=0$. Since $r=\ln z/\ln 2$, the roots are $-1,-2$. Thus $p=3$, $q=2$.

Writing $y=Ae^{-t}+Be^{-2t}$, the first two samples give $A+B=3$, $A/2+B/4=1$, so $A=1$, $B=2$. Therefore $$\boxed{y=e^{-t}+2e^{-2t}},\qquad \boxed{y'(0)=-5}.$$ The four sample values are $3$, $1/2+2/4=1$, $1/4+2/16=3/8$, and $1/8+2/64=5/32$, as required. Both coefficients are nonzero.

Step 4: Identify the missing-information case. For a pure observed mode $S_n=Cz^n$, the recurrence only imposes $z^2-sz+k=0$. Any positive second multiplier different from $z$ gives another pair $s,k$ with exactly the same samples. Thus the active root can be identified, but an unexcited root remains undetermined, even with arbitrarily many noiseless samples of that one trajectory. Exact data do not reveal a mode whose coefficient is zero.