Boundary Value Problems — Question 2

Question 2 – Solution

Strategy. Test whether the endpoint equations independently determine the homogeneous constants.

Step 1: Derive compatibility. The general solution is $y=1+A\cos x+B\sin x$. The left endpoint gives $A=\alpha-1$, while the right endpoint gives $\beta=1-A=2-\alpha$. Thus $$\boxed{\text{A solution exists exactly when }\alpha+\beta=2.}$$

Step 2: Count the compatible solutions. When compatible, the entire family is $$\boxed{y=1+(\alpha-1)\cos x+B\sin x,\qquad B\in\mathbb R.}$$ There are infinitely many solutions; otherwise there are none. There is never exactly one with just these two conditions. The nonzero homogeneous solution $\sin x$ vanishes at both endpoints and can be added without changing the data.

Step 3: Use an integral normalization. Since the cosine integrates to zero and the sine integrates to two, $$\int_0^\pi y\,dx=\pi+2B=m,\qquad \boxed{B=(m-\pi)/2.}$$ For compatible endpoints, every real $m$ selects exactly one solution. If the endpoints are incompatible, an extra integral condition cannot fix that.

Step 4: Change the interval and compare. On $[0,\pi/2]$, the right endpoint gives $B=\beta-1$ independently of $A=\alpha-1$, so every pair gives one solution. Independence of the endpoint equations, not their number alone, makes the difference. The graph on the original interval shows $y=1+B\sin x$ for $B=-1,0,1$.