Question

Solve:

$xdy+ydx+\frac{xdx+ydy}{x^2+y^2}=0$

Answer 1

Given the differential equation

$xdy+ydx+\frac{xdx+ydy}{x^2+y^2}=0$

on, $d\left(xy\right)+\frac{1}{2}\frac{d(x^2+y^2)}{x^2+y^2}=0$

Now integrating we have,

$xy+\frac{1}{2}(x^2+y^2)=c$ [Where ${}^{\prime }{e}^{\prime }$ is integrating constant ]

on, $2xy+(x^2+y^2)=2c$

on, $(x+y{)}^2=2x.$