Hello Prathik,

Hope everything going well.

Here is a method from which you can solve such equation:

Multiplying by 2x/x^2

2(xdx+ydy)+4y(xdy−ydx)/x^2=0

d(x^2+y^2)+4yd(y/x)=0

Converting to polar co ordinates

d(r^2)+4rsinθd(tanθ)=0

2rdr+4rsinθsec^2(θ)dθ=0

2rdr+4rsinθsec^2(θ)dθ=0

dr+2secθtanθdθ=0

Integrating,

r+2secθ=C

√( x^2+y^2)+2 √( 1+tan^2(θ))=C

√( x^2+y^2)+2 √( 1+(y^2/x^2))=C

(x+2) √(x^2+y^2)=Cx

Hope this helps you. Thank you.

All the best!!