1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Homogeneous Differential Equation

  1. Sep 8, 2011 #1
    1. The problem statement, all variables and given/known data

    Find the general solution of the differential equation:


    2. Relevant equations

    I want to solve this as a homogeneous differential equation, so our equations are:
    v=[itex]\frac{y}{x}[/itex], y=vx, [itex]\frac{dy}{dx}[/itex]=v+x[itex]\frac{dv}{dx}[/itex]

    3. The attempt at a solution

    I need to get this into the form [itex]\frac{dy}{dx}[/itex]=F([itex]\frac{y}{x}[/itex]), so I rewrite it as y'=[itex]\frac{x-y}{x+y}[/itex]. Dividing by x I get, y'=[itex]\frac{1-\frac{y}{x}}{1+\frac{y}{x}}[/itex]. From here, I substitute to get v+x[itex]\frac{dv}{dx}[/itex]=[itex]\frac{1-v}{1+v}[/itex]. When looking at the solution manual, however, it says that it should be in the form x(v+1)v'=-(v2+2v-1) before I integrate. I don't see how i can get it into this form. Also, it gives the answer as y2+2xy-x2=C. I can correctly solve this differential equation using different methods, but I would really like to know how to solve this using the homogeneous method. I would really appreciate any help with this. Thanks!
  2. jcsd
  3. Sep 8, 2011 #2


    User Avatar
    Homework Helper

    You are almost there...

    You have to compute:
    This will be a separable equation which is solved in the usual fashion.
  4. Sep 8, 2011 #3
    you just have to take subtract v on both sides of the equation you got man......(xdv/dx=(1-v)/(1+v) - v..
  5. Sep 8, 2011 #4
    Ok, so I rewrite it as [itex]\frac{1}{x}[/itex]dx=[itex]\frac{1+v}{1-v}[/itex]-[itex]\frac{1}{v}[/itex]dv. Do I need to do an additional substitution on [itex]\frac{1+v}{1-v}[/itex]?
  6. Sep 8, 2011 #5
    you solved it wrong on right side... you will get dv/{(1-v)/(1+V) - v}....which comes out to be (1+v)/(1-v^2-2v)...just substitute 1-v^2-2v as t....
  7. Sep 8, 2011 #6
    I'm having trouble seeing how it comes out to (1+v)/(1-v2-2v)
  8. Sep 8, 2011 #7


    User Avatar
    Homework Helper

    Do the addition as I said to get:
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook