Problem $2$

Consider the differential equation

$\frac{dy}{dx}=x+siny$

and note that the second derivative of $y$ is given by $($no need to show your work$)$:

$\frac{d^{2}y}{d{x}^2}=1+xcosy+\frac{1}{2}sin2y$

Answer the following questions:

$($a$)$ A solution curve passes through the point $(1,\frac{}{2}).$ What is its slope at this point?

$($b$)$ Justify why every solution curve increases for $x>1.$

$($c$)$ A solution curve passes through $(0,0).$ Show that this curve has a relative minimum at $(0,0).$