How do I solve 3xy^3dy/dx=y^2x^2+5x^3y+y^4

Answer 1

Let u=y/x, then y=u*x and y'=u'*x+u

We have the equation:
3x(u*x)^3(u'x+u)=(u*x)^2x^2+5x^3(u*x)+
+(u*x)^4
Divide on u*x^4
3u^2(u'x+u)=u+5+u^3
u'x+u=(u^3+u+5)/3u^2
u'x=(-2u^3+u+5)/3u^2
3u^2du/(-2u^3+u+5)=dx/x (u'=du/dx)