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Math Help - Making a substitution in a differential equation.

  1. #1
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    Post Making a substitution in a differential equation.

    The variables x and y are related by the differential equation:

    \frac{dy}{dx} = \frac{x^2 - y^2}{xy}

    Use the substitution y=xz, where z is a function of x, to obtain the differential equation:

    x\frac{dz}{dx} = \frac{1-az^2}{z}

    My questions sheet has 2 instead of the a, but I get a=1, which is correct?

    Cheers.
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  2. #2
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    Quote Originally Posted by Nyoxis View Post
    The variables x and y are related by the differential equation:

    \frac{dy}{dx} = \frac{x^2 - y^2}{xy}

    Use the substitution y=xz, where z is a function of x, to obtain the differential equation:

    x\frac{dz}{dx} = \frac{1-az^2}{z}

    My questions sheet has 2 instead of the a, but I get a=1, which is correct?

    Cheers.
    The a should be a 2.

    x \frac{dz}{dx} + z = \frac{x^2 - x^2z^2}{x^2z}

    and isloating z' gives

    x \frac{dz}{dx} = \frac{1 - 2z^2}{z}.
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  3. #3
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    Quote Originally Posted by danny arrigo View Post
    The a should be a 2.

    x \frac{dz}{dx} + z = \frac{x^2 - x^2z^2}{x^2z}
    Ah I see where I went wrong, I forgot to use the product rule when differentiating y=xz, thanks for the help.
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