II \(\displaystyle \frac{dy}{dx}=\frac{-(x+1)}{y- 2}\) is the same as (y- 2)dy= -(x+1)dx.
Integrate!
However you will not get the result shown! It should be \(\displaystyle (y- 2)^2\), not \(\displaystyle (y-1)^2\).
III Do you know what a "direction field" is? At each (t, y) point draw a short line segment with slope 5- 2y. That does not depend upon "t" so, for example, at each point on the y= 0 axis, (t, 0), you will have short line with slope 5, pretty steep. At each point on the y= 1 line, (t, 1) you will have short line with slope 3, y= 2, slope 1, y= =3, -1, etc.