1. Not finding help here? Sign up for a free 30min 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!

Solve Homogeneous D.E. need help integrating

  1. Mar 5, 2014 #1
    1. The problem statement, all variables and given/known data

    Dy/Dx = (Y-x)/(Y+x)

    2. Relevant equations

    Y=ux
    dy=udx+xdu



    3. The attempt at a solution

    Dy/Dx = (Y-x)/(Y+x)

    Plug in my substitutions
    udx+xdu(1/dx)=(ux/ux+x) - X/(ux+x)

    Simplify
    u+x(du/dx)=(ux)/x(u+1) - (x)/((x)(u+1))

    u+x(du/dx)=u/(u+1) -(1)/(u+1))

    u+x(du/dx)=u-1/(u+1)

    This is where I think i begin to mess up

    u+du=(u-1)/(u+1) dx/x

    substract (u-1)/(u+1) to the other side

    u-(u-1)/(u+1) du=dx/x

    I know the right side integrates to Lnx +c

    but on the left side if i do
    (u^2-1)/(u+1)

    I split it up into

    the integral (u^2)/(u+1) minus integral of 1/(u+1)
    (u^2)/(u+1)<-use long division

    I get u+(1/u+1) minus the integral of 1/(u+1)
    i am left with just the integral
    of u

    u^2/2= lnx+c

    plug u back in.

    ((y/x)^2)/2 =lnx +c

    is this sufficient of an answer?



    according to the answer key im gonna end up with the arctan somewhere in my answer. so i may have already messed up :(
     
    Last edited: Mar 6, 2014
  2. jcsd
  3. Mar 6, 2014 #2

    pasmith

    User Avatar
    Homework Helper

    Why are you dividing by [itex]y + x[/itex] here? Is your equation actually
    [tex]
    \frac{dy}{dx} = \frac{y - x}{y + x}
    [/tex]
    and not
    [tex]
    \frac{dy}{dx} = (y - x)(y + x)
    [/tex]
    as you have written?

    This should be "u dx/x + du" on the left hand side.

    You can't; it's multiplied by dx/x.

    What you have after replacing [itex]y[/itex] is
    [tex]
    x\frac{du}{dx} + u = \frac{u - 1}{u + 1}
    [/tex]
    Subtracting [itex]u[/itex] from both sides and then dividing by [itex]x[/itex] puts this in the separable form
    [tex]
    \frac{du}{dx} = \frac1x \left(\frac{u-1}{u+1} - u\right)
    [/tex]
    Continue.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Solve Homogeneous D.E. need help integrating
Loading...