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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
Evaluate the Differential Equation
Answer
The given differential equation is
This means that the slope of the solution curve, represented by ( \frac{dy}{dx} ), depends on the value of ( y ). Since ( \sin(y) ) oscillates between -1 and 1, the right-hand side will oscillate between 0 and 2. Thus, ( \frac{dy}{dx} \geq 0 ) for all ( y ), indicating that the function is always increasing.
Step 2
Analyze the Graph Options
Answer
We need to choose the graph that represents a function that is always increasing:
- Option A: This graph decreases at some points; hence it can't be the solution.
- Option B: This graph approaches a horizontal line as x approaches infinity, which is consistent with an increasing function that approaches a limit.
- Option C: Similar to option A, it demonstrates regions of decreasing behavior.
- Option D: This graph also has regions where it decreases, disqualifying it.
Thus, option B is the only one that fits the criteria for being a solution to the differential equation.
Step 3