Definitions — Question 7

Question 7 – Solution

Strategy. Translate each condition into an equation for $A$ and $B$. Conditions may be independent, redundant, or inconsistent.

Step 1: Verify the family. We have $$y'=-A\sin x+B\cos x,\qquad y''=-A\cos x-B\sin x=-y.$$ Thus every family member solves $y''+y=0$ on $\mathbb R$.

Step 2: Initial conditions. In (I), both values are prescribed at $x=0$, so this is an IVP. Since $y(0)=A$ and $y'(0)=B$, $$A=2,\quad B=-1,\qquad \boxed{y=2\cos x-\sin x.}$$ The supplied completeness of the family makes this the unique solution.

Step 3: Boundary conditions. In (II) and (III), values are prescribed at two different points, so these are BVPs. In both, $y(0)=2$ gives $A=2$. But $$y(\pi)=A\cos\pi+B\sin\pi=-A=-2,$$ independently of $B$. Consequently, $$\boxed{\text{(II): }y=2\cos x+B\sin x,\quad B\in\mathbb R}$$ has infinitely many solutions. In (III), the demand $y(\pi)=0$ contradicts $y(\pi)=-2$, so

Step 4: Why counting fails. Two written conditions need not provide two independent, consistent constraints. In (II) the second repeats information already forced by the first; in (III) it conflicts with that information. Their number alone cannot establish uniqueness or even existence.