Boundary Value Problems — Question 8

Question 8 – Solution

Strategy. Solve the differential equation completely, then retain the nonlinear structure of its endpoint constraint.

Step 1: Classify the slope equation. Every solution of $y''=0,y(0)=0$ is $y=cx$. Its remaining boundary condition is $c=c^2+\lambda$, so $$\boxed{c_\pm=\frac{1\pm\sqrt{1-4\lambda}}2.}$$ There are two real solutions for $\lambda<1/4$, one for $\lambda=1/4$, and none for $\lambda>1/4$. The critical solution is $y_*=x/2$.

Step 2: Check the superposition claim. At $\lambda=0$, the solutions are $0$ and $x$. Their average $x/2$ has $y'(1)=1/2$ but $y(1)^2=1/4$, so it fails the boundary condition. The differential equation is linear, but the square in the boundary operator makes the full boundary-value problem nonlinear.

Step 3: Quantify critical sensitivity. At $\lambda=1/4-\varepsilon$, the slopes are $1/2\pm\sqrt\varepsilon$. Hence $$\boxed{\max_{[0,1]}|y_\pm-y_*|=\sqrt\varepsilon.}$$ A Lipschitz bound with fixed constant $K$ would require $\sqrt\varepsilon\leq K\varepsilon$ for all sufficiently small positive $\varepsilon$, which is impossible. Both branches approach $y_*$ continuously from this side, but not at a rate bounded linearly by the parameter error.

Step 4: Read the complete branch picture. The branches meet at $(\lambda,c)=(1/4,1/2)$ and have no real continuation for larger $\lambda$. For $\lambda<1/4$, $dc_\pm/d\lambda=\mp 1/\sqrt{1-4\lambda}$, whose magnitude diverges at the join. Critical uniqueness therefore does not guarantee a locally unique family or a Lipschitz-stable response.