Problem 3. Consider the following initial value problem
$\frac{dy}{dx} = -\sqrt{1 - y^2}$, $y(a) = b$,
where $a$ and $b$ are real numbers. Answer the following questions.
(1) Let $c \in \mathbb{R}$ and define
$y_c(x) = \begin{cases} 1, & x \le c, \
\cos(x - c), & c < x < c + \pi, \
-1, & x \ge c + \pi.
\end{cases}$
(1)
Prove that $y_c(x)$ is a solution of $y' = -\sqrt{1 - y^2}$ for every $c$ and every $x \in \mathbb{R}$.
(2) Sketch a variety (choose a few values for $c$) of solution curves $y_c(x)$.
(3) Determine in terms of $a$ and $b$ how many different (local) solutions the
initial value problem (1) has.
(4) Explicitly write a unique local solution for (1), if any.
