Solution of the differential equation $\frac{x d y}{x^2+y^2}=\left(\frac{y}{x^2+y^2}-1\right) d x$, is

$\tan ^{-1}\left(\frac{y}{x}\right)=-x+C$

The given differential equation can be written as

$\frac{x d y-y d x}{x^2+y^2}=-d x \Rightarrow d\left\{\tan ^{-1}\left(\frac{y}{x}\right)\right\}=-d x$

On integrating, we obtain

$\tan ^{-1}\left(\frac{y}{x}\right)=-x+f C$, which is the required solution.