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Math Help - First Order ODE using homogeneous substituion

  1. #1
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    First Order ODE using homogeneous substituion

    someone please enlighten me on each step taken (as well as integrating both sides of the separable equation involving z and x) in solving the following equation:

    (x^2)y'=7(y^2)-2xy
    using the subsitution z=(y/x)

    i know what has to be done, but im having troubles simplifying into a regular funtion y(x)...

    this can be done using bernouilli's method but wanna get the same using the homogeneous substituion method

    thanks
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  2. #2
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    Krizalid's Avatar
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    y' = 7 \cdot \left( {\frac{y}<br />
{x}} \right)^2  - 2 \cdot \left( {\frac{y}<br />
{x}} \right).

    Now make y=xz.
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  3. #3
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    yea i did that, but solve down to the part where you end up with a separable equation dz/7(z^2)-3z = dx/x.

    i tried integrating both sides, and especially the left one needed more tactics where i used partial integration {dz/7(z^2)-3z} = 1/7[A/z + B/{z-(3/7))]

    my problem here is integrating and getting a solution y(x) after back substituting z=(y/x). I did the same ODE using bernouillis and i got
    y(x)=-7x^(-1) +Cx^(-2). Just want to see how to reach the same results using the homogenous substitution technique

    thanks in advance!
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