Boundary Value Problems — Question 3

Question 3 – Solution

Strategy. Integrate the flux balance before trying to determine the additive level of the response.

Step 1: Check the interval balance. The equation is $q'=2x-1$, so $$q(x)=q_0+x^2-x,\qquad q_1-q_0=\int_0^1(2x-1)\,dx=0.$$ Thus $\boxed{q_1=q_0}$ is necessary. It is sufficient because division by $1+x>0$ and one integration produce a solution for every such pair.

Step 2: Solve the zero-flux case. Here $y'=(x-x^2)/(1+x)=-x+2-2/(1+x)$, hence $$\boxed{y=C-x^2/2+2x-2\log(1+x).}$$ Multiplication by $-(1+x)$ gives $q=x^2-x$, whose derivative is $2x-1$ and whose values at zero and one are zero. This checks the equation and flux data.

Step 3: Fix the additive constant. Using $\int_0^1\log(1+x)\,dx=2\log 2-1$, we obtain $$\int_0^1y\,dx=C+\frac{17}{6}-4\log 2.$$ Thus $\boxed{C=4\log 2-17/6}$. The difference of two solutions with the same fluxes has $((1+x)w')'=0$ and zero endpoint flux, so $w'=0$. It is constant; the zero-mean condition forces that constant to vanish.

Step 4: Interpret the resulting profile. Since $y'=x(1-x)/(1+x)>0$ for $0<x<1$, the response is strictly increasing. Its endpoint values are $$\boxed{y(0)=4\log 2-17/6,\qquad y(1)=2\log 2-4/3.}$$ Fluxes involve derivatives and therefore leave a constant level undetermined. Adding a constant changes neither flux nor the integrated balance, so it cannot repair a violation of $q_1=q_0$.