Which of the following could be the graph of a solution to the differential equation dy/dx = sin y + 1? A - HSC - SSCE Mathematics Extension 1 - Question 10 - 2022 - Paper 1

Notice that $\sin(y)$ oscillates between -1 and 1. Thus, we have:

$\frac{dy}{dx} = \sin(y) + 1$

This results in:

• When $\sin(y) = -1$, then $\frac{dy}{dx} = 0$.
• When $\sin(y) = 0$, then $\frac{dy}{dx} = 1$.
• When $\sin(y) = 1$, then $\frac{dy}{dx} = 2$.

This indicates that $\frac{dy}{dx}$ will always be non-negative, suggesting that the graph will never slope downwards.

The critical point occurs when $\frac{dy}{dx} = 0$. This happens when:

$\sin(y) + 1 = 0 \Rightarrow \sin(y) = -1 \Rightarrow y = \frac{3\pi}{2} + 2k\pi, k \in \mathbb{Z}$

Therefore, $y$ has local maxima at these points while the function increases between them.

Given that $\frac{dy}{dx}$ is always non-negative, the graph will consistently either rise or maintain a constant value, but never decrease. Among the options listed, the graph that correctly represents this behavior is B, which shows a general increase and approaches a horizontal asymptote, indicating no further increase in $y$ as $x$ goes to infinity.