Consider the differential equation \( \frac{dy}{dx} = \frac{x}{y} \) - HSC - SSCE Mathematics Extension 1 - Question 4 - 2021 - Paper 1

Integrating both sides gives us:

$\int y \, dy = \int x \, dx$

This results in:

$\frac{y^2}{2} = \frac{x^2}{2} + C$, where C is the constant of integration.

To express this in standard form, we multiply through by 2 to eliminate the fractions:

$y^2 = x^2 + 2C$

We can define a new constant ( c = 2C ) to simplify the equation further as:

$y^2 = x^2 + c$.

From the derived equation ( y^2 = x^2 + c ), we see that option A, ( y^2 = \frac{x^2}{2} + c ), is the closest match, conflicting slightly with our derived constants. Therefore, option A best represents this relationship.