The differential equation $x \frac{\mathrm{dy}}{\mathrm{d} x}=2 \mathrm{y}$ represents ....

The differential equation $x\frac{dy}{dx}=2y$ separates as $\frac{dy}{y}=\frac{2}{x}dx$.

Integration yields $\ln|y|=2\ln|x|+C_1$, which simplifies to $\ln|y|=\ln|x^2|+C_1$. Writing $C_1=\ln|C|$ gives $\ln|y|=\ln|Cx^2|$, so exponentiating results in $|y|=|Cx^2|$.

This represents the family of curves $y=Cx^2$, where $C$ is an arbitrary constant.

These are parabolas with vertices at the origin and axes along the Y-axis.

Comparing options:

A: Family of circles (incorrect)

B: Family of parabolas with vertex at origin along positive Y-axis (correct)

C: Family with axis along X-axis (incorrect)

D: Family of ellipses (incorrect)