How can I solve this differential equation? : x y d x -...

we have in differential form:
${\displaystyle xy dx-(x^2+1) dy=0}$
If we put in standard form, and collect terms:
${\displaystyle \frac{1}{y}y \frac{dy}{dx}=\frac{x}{x^2+1}}$
Which is a First Order Separable Ordinary Differential Equation, so we can separate the variables to get:
${\displaystyle \int \frac{1}{y} dy=\int \frac{x}{x^2+1} dx}$
We can manipulate the RHS integral as follows:
${\displaystyle \int \frac{1}{y} dy=\frac{1}{2} \int \frac{2x}{x^2+1} dx}$
And now both integrals are standard results, so integrating give us:
${\displaystyle \ln \left|y\right|=\frac{1}{2}\ln |x^2+1|+C}$
Noting that we require areal solution, and writing ${\displaystyle C=\ln A}$, we get:
${\displaystyle \ln y=\ln A\sqrt{x^2+1}}$
Giving us the General Solution:
${\displaystyle y=A\sqrt{x^2+1}}$