Final Thoughts — Question 4

Question 4 – Solution

Strategy. Check the function’s domain as well as the equation’s right-hand side. A solution can become unbounded in finite time even for a smooth equation.

Step 1: Verification and interval. For $x\ne 1$, $$f'(x)=\frac 1{(1-x)^2}=f(x)^2,\qquad f(0)=1.$$ The largest open interval containing $0$ that avoids $1$ is $\boxed{(-\infty,1)}$.

Step 2: No finite extension. The limits are $$\boxed{\lim_{x\to 1^-}f(x)=+\infty,\qquad \lim_{x\to 1^+}f(x)=-\infty.}$$ A differentiable extension across $1$ would be continuous there and would have a finite value. The left-hand divergence rules this out, independently of what formula is proposed after $1$.

Step 3: The separate branch. On $(1,\infty)$, the same derivative calculation still proves $f'=f^2$. This branch is a solution of the differential equation, but its interval does not contain the initial point $0$ and cannot be joined through $x=1$ to the IVP branch.

Step 4: Corrected inference. Smoothness of a right-hand side is not a guarantee of existence for all time. Here the equation is defined at every finite pair $(x,y)$, but the IVP solution escapes to unbounded $y$ as $x\to 1^-$. The failure belongs to global continuation of the solution, not to a singularity of $y^2$ at a finite point.