ODE's

Discussion in 'Math' started by Hello, Apr 16, 2009.

  1. Hello

    Thread Starter Active Member

    Dec 18, 2008
    82
    0
    Using the substitution y = 1/z, find the solution of the following differential equation:

    dy/dx = y + x(y^2)



    This is what i've done so far:

    dy/dx = (dy/dz).(dz/dx)

    Not sure what to do next or if it is correct!

    Any help would be greatly appreciated.
     
    Last edited: Apr 16, 2009
  2. mattc82

    Member

    Mar 13, 2009
    22
    0
    I'm not 100% but I would seperate and integrate in y and x first,then sub y=1/z after. Unless this is supposed to be PDE?
     
  3. steveb

    Senior Member

    Jul 3, 2008
    2,433
    469
    That is a good start. Now complete the substitution in the entire equation. The dy/dz can be easily identified from y=1/z (i.e. dy/dz=-1/z^2)

    Once you make the substitution you will see that the final equation is in a very simple form.

    dz/dx=-z-x

    This is a simple first order system with a negative ramp (-x) as the input function.
     
  4. Ratch

    New Member

    Mar 20, 2007
    1,068
    3
    Hello,

    dy/dx = (-1/z^2)dz/dx

    so (-1/z^2)dz/dx = 1/z + x/z^2 ==> -dz/dx = z + x ==> -dz = zdx + xdx

    dz + zdx = -xdx ==> (e^x)dz + z(e^x)dx = -x(e^x)dx ==> d((e^x)z) = d((e^x-x(e^x)) ==> (e^x)z = (e^x-x(e^x) +C ==> z = 1-x+C/e^x
    ==> 1/y = 1-x+C/e^x ==> y = 1/(1-x+C/e^x)

    Ratch
     
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