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ODE: Confused about a Homogeneous Eqn question

  • Thread starter hahutzy
  • Start date
  • #1
10
0

Homework Statement



So the problem goes like this:

Code:
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

Homework Equations



(Included)

The Attempt at a Solution



Code:
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??
 

Answers and Replies

  • #2
[tex] \frac {dy}{dx} = x \frac {dv}{dx} + v [/tex]
 
  • #3
10
0
[tex] \frac {dy}{dx} = x \frac {dv}{dx} + v [/tex]
Thank you!
 

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