Solve differential equation by an appropriate substitution

(x^2 + 3xy + y^2) dx − x^2 dy = 0

Please explain.

Solve for dy/dx:

dy/dx = (x^2 + 3xy + y^2) / x^2 = 1 + 3(y/x) + (y/x)^2

This has the form dy/dx = f(y/x), so this DE is homogeneous of degree 1.

The substitution is u = y/x.

y = ux
dy/dx = x du/dx + u

Making the substitution,

x du/dx + u = 1 + 3u + u^2
x du/dx = 1 + 2u + u^2 = (1 + u)^2
[1/(1+u)^2] du = (1/x) dx

Integrate both sides:

-1/(1+u) = ln(x) + c

Replace u by y/x:

-1/(1 + (y/x)) = ln(x) + c

I'll let you finish.

Make ur quest clearer either u differentiate implicitly 2 get dy/dx=-3y-2x/3x-2y or you differentiate both with respect to dx only to get 2x-2x=0

Use separation of variables. I come up with sqrt(3x squared) = y
I may not be right, check a Diff Eq book