Basic Concepts — Question 4

Question 4 – Solution

Strategy. Translate each condition into a constraint on $A,B$ and check whether the second constraint supplies new information.

Step 1: Use value and slope at the same point. From $y(0)=A$ and $y'(0)=B$, initial data $0,b$ give $$\boxed{y=b\sin t}.$$ Both constants are fixed, for every real $b$. The coefficients of the normalized equation are continuous everywhere, consistent with initial-value uniqueness.

Step 2: Move the second datum to $\pi/2$. The first condition gives $A=0$. At the other endpoint, $$y(\pi/2)=B=c, \qquad \boxed{y=c\sin t}.$$ This particular pair of endpoint conditions also determines one solution.

Step 3: Detect a redundant or incompatible condition. At $L=\pi$, every function $B\sin t$ has $y(\pi)=0$. Consequently, $$\boxed{c=0:\ y=B\sin t\ (B\in\mathbb R)},\qquad \boxed{c\ne 0:\ \text{no solution}}.$$ When $c=0$, the second condition adds no restriction on $B$. When $c\ne 0$, it contradicts the first condition and the equation. Merely counting two stated conditions misses this distinction.

Step 4: Compare endpoint agreement with initial-data agreement. The four curves share their endpoint positions but have different initial slopes $B$. At $t=\pi$ their slopes are $-B$, also different. The initial-value theorem requires agreement of both value and first derivative at the same point. Agreement of values at two separate points is a different requirement, so the sketch creates no contradiction.