Photo AI
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
Question 10
Which of the following could be the graph of a solution to the differential equation dy/dx = sin(y) + 1? A. B. C. D.
Worked Solution & Example Answer: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
Step 1
Analyze the differential equation
Answer
The given differential equation is of the form This indicates that the rate of change of y with respect to x is determined by the sine of y plus a constant 1. Notably, since sin(y) oscillates between -1 and 1, it follows that (\sin(y) + 1) oscillates between 0 and 2. Therefore, (\frac{dy}{dx} \geq 0) indicating that y is always non-decreasing.
Step 2
Consider the implications for the graph
Answer
Given that (\frac{dy}{dx} \geq 0), the graph must either be increasing or remain constant. This rules out any options that display a decreasing trend as they would imply (\frac{dy}{dx} < 0). Furthermore, since the minimum value of (\frac{dy}{dx}) is 0 (when (\sin(y) = -1)), the graph will never have horizontal lines that collapse to a single point, ensuring that the function is always rising or leveling off, never declining.
Step 3
Evaluate each option
Answer
Considering the options given:
- Option A shows a decreasing function, thus invalid.
- Option B shows a flat line at a certain y-value for large x, suggesting a meaningful upward trend, making it a valid solution.
- Option C displays oscillations that extend down, which is not consistent with a non-decreasing function.
- Option D shows a general upward trend with oscillations but may not meet the non-decreasing requirement overall.
Thus, Option B is the only plausible option as it represents a function that continues to rise or level out without decreasing.
Step 4