Consider the differential equation \( \frac{dy}{dx} = \frac{x}{y} \).
Which of the following equations best represents this relationship between x and y ?
A. \( y^... show full transcript
Worked Solution & Example Answer:Consider the differential equation \( \frac{dy}{dx} = \frac{x}{y} \) - HSC - SSCE Mathematics Extension 1 - Question 4 - 2021 - Paper 1
Step 1
Differentiate the equation
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Answer
Starting with the differential equation ( \frac{dy}{dx} = \frac{x}{y} ), we can multiply both sides by ( y ) to separate the variables. This gives us ( y , dy = x , dx ).
Step 2
Integrate both sides
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Answer
Next, we integrate both sides:
[ \int y , dy = \int x , dx ]
This simplifies to
[ \frac{y^{2}}{2} = \frac{x^{2}}{2} + c ]
where ( c ) is the constant of integration.
Step 3
Rearranging the equation
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Answer
Rearranging this yields
[ y^{2} = x^{2} + 2c ]
Since ( 2c ) is still a constant, we can denote it as a new constant ( c' ) and rewrite it as
[ y^{2} = x^{2} + c' ], where ( c' = 2c ).
Step 4
Select the correct answer
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Answer
Given the options, the equation ( y^{2} = x^{2} + c ) matches our derived equation best. Therefore, the correct option is A.
