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

The term $\sin y + 1$ varies between 1 and 2 because the sine function oscillates between -1 and 1. This indicates that $\frac{dy}{dx}$ is always positive, implying that the graph of y increases as x increases.

We need to determine which graph shows a continuously increasing function.

• Option A: This graph has turning points, indicating that the function decreases at some intervals, which is inconsistent with $\frac{dy}{dx} > 0$.

• Option B: This graph is a horizontal line after a certain point, suggesting that the function stabilizes and stops changing; however, it matches the behavior of an asymptotic solution.

• Option C: Similar to Option A, it exhibits decreasing behavior, which is not permissible for our differential equation.

• Option D: This exhibits oscillatory behavior indicating regions of decrease, which contradicts the always positive derivative.

Based on this analysis, Option B is the only one that can represent the function governed by our differential equation.

The correct answer is Option B, as it shows a function that stabilizes after increasing, aligning with the behavior dictated by the differential equation.