Solve the given differential equation by separation of varia

Solve the given differential equation by separation of variables. y ln xdx/dy = (y+1/x)^2 y^2/2+2y+ln|y|=

x*ln(x)*dx = (y+1)^2*dy/y

(y^2 + 2y + 1) / y * dy = x*ln(x)*dx

(y + 2 + 1/y) * dy = x^2*ln(x)*dx

Integrating both sides :

y^2/2 + 2y + ln|y| = integral of (x^2*ln(x)*dx)

USing parts :

u = ln(x) , dv = x^2
du = 1/x , v = x^3/3

uv - (integral) vdu

x^3*ln(x)/3 - (integral) x^2/3

x^3*ln(x)/3 - x^3/9 + C

So the solution is :

y^2/2 + 2y + ln|y| = x^3*ln(x)/3 - x^3/9 + C ---> ANSWER