Question

note: if you are asked for value(s), give your answer as a single value (e.g. 17), or a comma separated list (e.g. 1,2,3), or none.
if you are asked for an interval, give your answer in helio (intervals).
type inf for \\( \infty \\) and -inf for \\( -\infty \\).
consider the slope field below for a differential equation \\( \frac{d y}{d t}=f(y) \\) with \\( y\left(t_0\
ight)=y_0 \\).
(a) determine all \\( y_0 \\) values such that the solution to the differential equation is constant (separated by commas, if applicable). \\( y_0= \\)
(b) if \\( y_0=-1 \\), determine \\( \lim _{t \
ightarrow \infty} y(t)= \\)
(c) determine all values of \\( y_0 \\) such that \\( \lim _{t \
ightarrow \infty} y(t) \\) is a positive real number (use interval notation). \\( y_0 \in \\)

Explanation:

Step1: Analyze constant solutions

For a constant solution \(y(t) = y_0\), we substitute into \(\frac{dy}{dt}=f(y)\). A constant solution satisfies \(\frac{dy}{dt} = 0\). Looking at the slope - field, the slopes are zero (horizontal line segments) at \(y = 3\), \(y = 0\), and \(y=-2\).

Step2: Analyze the limit as \(t

ightarrow\infty\) for \(y_0=-1\)
When \(y_0=-1\), we observe the direction of the slope - field. As \(t\) increases, the solution \(y(t)\) approaches \(y = 0\)

Step3: Analyze the limit as \(t

ightarrow\infty\) for positive real number
We look at the regions of the slope - field. If \(y_0>0\), as \(t
ightarrow\infty\), \(y(t)\) approaches \(y = 3\). So the interval for \(y_0\) such that \(\lim_{t
ightarrow\infty}y(t)\) is a positive real number is \((0,\infty)\)

Answer:

(a) \(y_0 = 3,0,-2\)
(b) \(0\)
(c) \(y_0\in(0,\infty)\)