Undetermined Coefficients — Question 7

Question 7 – Solution

Strategy. Fit the initial data before taking the frequency limit; the homogeneous correction cancels the divergent constant part.

Step 1: Solve off resonance. The trial $A\cos\omega t$ gives $A(1-\omega^2)=1$. Fitting zero data adds $-\cos t/(1-\omega^2)$, so $$\boxed{y_\omega(t)=\frac{\cos(\omega t)-\cos t}{1-\omega^2},\qquad\omega\ne 1.}$$ Both the value and slope at zero vanish.

Step 2: Solve at resonance. Use $t(A\cos t+B\sin t)$. Applying $D^2+1$ gives $-2A\sin t+2B\cos t$, hence $A=0$, $B=1/2$. Its data are already zero, so $\boxed{y_1=(t/2)\sin t}$.

Step 3: Take the fixed-time limit. Differentiating numerator and denominator with respect to $\omega$ gives $$\boxed{\lim_{\omega\to 1}y_\omega(t)= \frac{-t\sin t}{-2}=\tfrac 12t\sin t.}$$ This includes $t=0$. The particular coefficient and the cosine correction separately diverge, but their full numerator vanishes at the same rate as the denominator. They must be combined before taking the limit.

Step 4: Compare long-time behavior. For fixed $\omega\ne 1$, $|y_\omega|\le 2/|1-\omega^2|$ on the whole half-line. The resonant response is unbounded, for example at $t=\pi/2+2\pi n$. Consequently, for each fixed nonresonant $\omega$, the difference $y_\omega-y_1$ is unbounded on that half-line. Uniform convergence there is impossible, although the fixed-time limit exists. A finite-window plot cannot replace this distinction.