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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.

          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.

Problem 3. Consider the following initial value problem
(dy)/(dx) = -√(1 - y^2), y(a) = b,
where a and b are real numbers. Answer the following questions.
(1) Let c ∈ℝ and define
yc(x) = 1, x ≤ c, cos(x - c), c < x < c + π, -1, x ≥ c + π.
(1)
Prove that yc(x) is a solution of y' = -√(1 - y^2) for every c and every x ∈ℝ.
(2) Sketch a variety (choose a few values for c) of solution curves yc(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.

Added by Michael F.

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