I found the first derivative as:
\frac{dy}{dx} + 0 = 1 \\ \\ \\ \therefore \frac{dy}{dx} = 1
So we get that result from y(0) = 0.
I found the implicit derivative with respect to x as:
\frac{d^2 y}{dx^2} + 2y \frac{dy}{dx} = -\sin (x)
Then I got that when we substitute y(0) = 0, we get...