[latex] \[
\textbf{Q1: Solve the differential equation:} \quad x \, dx + 2xy \, dy = 0
\]
\[
\textbf{Solution:}
\]
\[
x \, dx = -2xy \, dy
\]
\[
\frac{dx}{dy} = -2y
\]
Now integrate both sides:
\[
\int 1 \, dx = \int -2y \, dy
\]
\[
x = -y^2 + C
\]
Thus, the solution to the differential equation is:
\[
x = -y^2 + C
\]