Consider the first and last to get one equation and then consider $\dfrac{\mathrm{d} y}{y} = \dfrac{\mathrm{d} u + \mathrm{d} x}{x + u}$
Solution: Let us write the auxiliary equation.
\begin{equation}\label{eq:11Apr2025-1}
\frac{\mathrm{d} x}{u} = \frac{\mathrm{d} y}{y} = \frac{\mathrm{d} u}{x}.
\end{equation}
Then considering the first and the last, we get
\begin{align*}
\frac{\mathrm{d} x}{u} = \frac{\mathrm{d} u}{x} & \implies x \mathrm{d} x = u \mathrm{d} u \\
& \implies \frac{x^2}{2} = \frac{u^2}{2} + c_1 \\
& \implies u^2 - x^2 = C_1,
\end{align*}
where $c_1$ is the constant of integration.
We now consider,
\begin{align*}
\frac{\mathrm{d} y}{y} = \frac{\mathrm{d} u + \mathrm{d} x}{x + u} & \implies \frac{\mathrm{d} y}{y} = \frac{d(u + x)}{u + x} \\
& \implies \ln y = \ln (x + u) + \ln (c_2) \\
& \implies \frac{y}{x+u} = C_2,
\end{align*}
where $c_2$ is the constant of integration.
Thus, the solution of the equation \eqref{eq:11Apr2025-1} will be
\[
f\left( u^2 - x^2, \frac{y}{x + u} \right) = 0,
\]
which is option (A). Note that option (B) is also correct, as
\begin{align*}
u^2 - x^2 = g\left( \frac{y}{x + u} \right) \implies u^2 = g\left( \frac{y}{x + u} \right) + x^2.
\end{align*}
Therefore, the correct options are (A) and (B).