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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
Identify the behavior of the differential equation
Answer
The given differential equation is dy/dx = sin(y) + 1. The right-hand side, sin(y) + 1, takes values between 0 and 2 because sin(y) oscillates between -1 and 1. This indicates that the derivative dy/dx is non-negative (always 0 or positive), meaning that the function y will be non-decreasing.
Step 2
Analyze the options for possible graphs
Answer
Based on the above analysis, we can conclude:
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Option A: This graph has regions where it decreases, which is not consistent with dy/dx being non-negative. Thus, it cannot be the correct graph.
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Option B: This graph consistently increases and appears to approach a horizontal line. This matches our expectation since dy/dx is always positive. Thus, this could represent a possible solution.
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Option C: This graph has both increasing and flat regions, which again does not fit the requirement of dy/dx being non-negative everywhere. This makes it an incorrect choice.
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Option D: Similar to option C, this graph shows decreasing behavior, which is not permissible given our constraint. Thus, this option is also incorrect.
Step 3