Question

{"given":"We have a family of curves given by $x^2 + y^2 - 2ay = 0$, where $a$ is an arbitrary constant. We need to eliminate the arbitrary constant $a$ to form the differential equation. ","key_observation":"To eliminate the arbitrary constant, we differentiate the given equation with respect to $x$ and then use algebraic manipulation to eliminate $a$ between the original equation and its derivative. This is the standard method for finding differential equations of families of curves.","option_analysis":[{"label":"(A)","text":"$2(x^2 - y^2)y' = xy$","verdict":"incorrect","explanation":"This option has the correct terms $(x^2 - y^2)$ and $xy$ but includes an extra factor of 2 on the left side that doesn't appear in the correct derivation."},{"label":"(B)","text":"$2(x^2 + y^2)y' = xy$","verdict":"incorrect","explanation":"This option incorrectly has $(x^2 + y^2)$ instead of $(x^2 - y^2)$, and also has an incorrect factor of 2 and wrong right-hand side."},{"label":"(C)","text":"$(x^2 - y^2)y' = 2xy$","verdict":"correct","explanation":"Differentiating $x^2 + y^2 - 2ay = 0$ gives $2x + 2yy' - 2ay' = 0$. Solving for $a$: $a = \\frac{x + yy'}{y'}$. Substituting back into the original equation and simplifying yields $(x^2 - y^2)y' = 2xy$."},{"label":"(D)","text":"$(x^2 + y^2)y' = 2xy$","verdict":"incorrect","explanation":"This option incorrectly has $(x^2 + y^2)$ instead of $(x^2 - y^2)$ on the left side, which doesn't match the correct algebraic manipulation."}],"answer":"(C)","formula_steps":[]}