Question

can someone solve this differential equation pls?​

Answer 1

Let y+1=u⇒dydx=dudx

dudx=x+ux−u=1+ux1−ux

Let v=ux⇒u=vx⇒dudx=v+xdvdx

v+xdvdx=1+v1−v

xdvdx=1+v1−v−v=1+v21−v

1−v1+v2dv=1xdx

∫(11+v2−v1+v2)dv=∫1xdx

tan−1v−12ln(1+v2)=ln|x|+C

2tan−1v−ln(1+v2)=2ln|x|+C

Substitute back: v=ux=y+1x

2tan−1(y+1x)=ln(1+(y+1)2x2)+ln(x2)+C

2tan−1(y+1x)=ln(x2+(y+1)2)+C

2tan−1(y+1x)=ln(x2+y2+2y+1)+C