So the problem goes like this:
dy/dx = ( x^2 + xy + y^2 ) / x^2 a) Show that it is a homogeneous equation. b) Let v = y/x and express the eqn in x and v c) Solve for y
The Attempt at a Solution
a) dy/dx = ( x^2 + xy + y^2 ) / x^2 * [(1/xy) / (1/xy)] = [(x/y) + 1 + (y/x)] / (x/y) Since RHS is expressed only in terms of y/x, therefore it is homogeneous. b) v = y/x y = vx dy/dv = x dy/dx = (dy/dv) (dv/dx) = x (dv/dx) .'.dy/dx = [(x/y) + 1 + (y/x)] / (x/y) becomes x (dv/dx) = [(1/v) + 1 + v] / (1/v) = v^2 + v + 1 dx/x = dv/(v^2 + v + 1)
The problem is, when I solve by integrating both sides, I get some gibberish arctan (2v/sqrt(3) + 1)term.
The solution in the book and solved on wolfram alpha is:
arctan (y/x) - ln |x| = c
However, this answer suggests that the equation in b) must've been
dx/x = dv/(v^2 + 1) <- no v term
Which is clearly not the case...
What am I doing wrong??