Chapter 1.3, Problem 6E

Consider the differential equation $\frac{dy}{dx}=x+\sin y.$

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

b. Argue that every solution curve is increasing for $x>1.$

c. Show that the second derivative of every solution satisfies $\frac{d^{2}y}{d{x}^2}=1+xcos y+\frac{1}{2}\sin 2y$.

d. A solution curve passes through $\left(\mathrm{0,\; 0}\right).$ Prove that this curve has a relative minimum at $\left(0, 0\right).$