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Math Help - 1st order homogeneous diff. equation

  1. #1
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    1st order homogeneous diff. equation

    hi all,

    i have this differential equation to solve...

    i know it is homogeneous, so i can put z = \frac{y}{x}

    but i still can't solve this...

    y` = \frac{x^2-xy+y^2}{x^2}

    i get to:

    z`x + z = 1 - z + z^2

    where do i go from here?


    thanks
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  2. #2
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    Quote Originally Posted by vonflex1 View Post
    hi all,

    i have this differential equation to solve...

    i know it is homogeneous, so i can put z = \frac{y}{x}

    but i still can't solve this...

    y` = \frac{x^2-xy+y^2}{x^2}

    i get to:

    z`x + z = 1 - z + z^2

    where do i go from here?


    thanks
    Move the z to the right and separate.
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  3. #3
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    how do i separate it?

    there's that 1, so i can't divide by z...
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  4. #4
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    Quote Originally Posted by vonflex1 View Post
    how do i separate it?

    there's that 1, so i can't divide by z...
    Here's what I mean

     <br />
x \frac{dz}{dx} = z^2-2z+1 = (z-1)^2<br />

    so

     <br />
\frac{dz}{(z-1)^2} = \frac{dx}{x}.<br />
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  5. #5
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    i understand what you did there,

    and i will get ln(z-1)^2 = lnx

    but what next?

    how do i get back afterward back to y?
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  6. #6
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    Quote Originally Posted by Danny View Post
    Here's what I mean

     <br />
x \frac{dz}{dx} = z^2-2z+1 = (z-1)^2<br />

    so

     <br />
\frac{dz}{(z-1)^2} = \frac{dx}{x}.<br />
    Integrating gives

     <br />
\frac{-1}{z-1} = \ln x + c<br />
.

    Solve for z and then use your substitution.
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  7. #7
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    thanks, i get it now.
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