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

Rearranging the equation, we can separate the variables: ( y , dy = x , dx ).

Integrating both sides gives:

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

This results in:

$\frac{y^2}{2} = \frac{x^2}{2} + C$

where ( C ) is the constant of integration.

Multiplying through by 2, we arrive at:

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

We can denote ( 2C ) as a new constant ( c ), leading to:

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

Among the options provided, the equation that best represents this relationship is: A. ( y^2 = x^2 + c )