# ODE: Confused about a Homogeneous Eqn question

1. May 26, 2009

### hahutzy

1. The problem statement, all variables and given/known data

So the problem goes like this:

Code (Text):
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
2. Relevant equations

(Included)

3. The attempt at a solution

Code (Text):
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??

2. May 26, 2009

### Random Variable

$$\frac {dy}{dx} = x \frac {dv}{dx} + v$$

3. May 26, 2009

Thank you!