Cant solve diff-eq with substitution

[Ryoukomaru]

Use subs $$y=xv$$ to show that $$(x^2+y^2)+2xy\frac{dy}{dx}=0, x>0$$ is $$x^3+3xy^2=k$$ where k is a constant.

I played around with this at school and if memory serves me correct i got something similar to $$\frac{dx}{dv}=\frac{-3}{2xv}-\frac{1}{2}$$ and after that i decided i wasnt on the right path and stopped. Need a little help here !

 

[arildno]

x is to remain the independent variable; v(x) is the dependent variable which replaces y(x)

 

[stunner5000pt]

What is v? Is v a function or a constant??

 

[arildno]

What is v? Is v a function or a constant??

v is defined as the function:
$$v(x)=\frac{y(x)}{x}$$

 

[HallsofIvy]

If y= xv, then dy/dx= x dv/dx+ v. Put that into your equation and replace y by xv and see what happens.