Boundary Value Problems — Question 6

Question 6 – Solution

Strategy. Count the integration constants together with any unknown model parameter, and then solve the actual endpoint equations.

Step 1: Find compatibility for a fixed load. The left conditions give $y=qx^3/6+cx^2/2$. The right conditions require $a=q/6+c/2$ and $b=q/2+c$. Subtraction yields $$\boxed{b-2a=q/6.}$$ This condition is necessary and sufficient: if it holds, $c=2a-q/3$ gives the unique solution; otherwise no solution satisfies all the prescribed data.

Step 2: Recover an unknown load. When $q$ is free, the same two equations determine both $q$ and $c$: $$\boxed{q=6(b-2a),\qquad y=(b-2a)x^3+(3a-b)x^2.}$$ There is one solution for every pair $(a,b)$. Direct differentiation verifies the constant load and the two endpoint slopes and values.

Step 3: Check the displacement from rest to rest. For $a=1,b=0$, $$\boxed{q=-12,\qquad y=3x^2-2x^3.}$$ Its third derivative is $-12$, its endpoint values are zero and one, and $y'=6x(1-x)$ vanishes at both ends. Since this derivative is positive inside, the displacement is strictly increasing on $(0,1)$.

Step 4: Interpret the extra measurement. For $q=6,a=1$, compatibility requires $b-2=1$, so $\boxed{b=3}$ and $y=x^3$. Any other prescribed right slope is inconsistent with those fixed data. The left conditions already remove two of the three integration constants; with fixed $q$, the remaining constant cannot fit arbitrary two right data. When $q$ is unknown, it supplies one more adjustable quantity, which the fourth measurement identifies. Counting conditions is useful only with this distinction.