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26 September, 16:45

Consider the differential equation dy/dx = (y - y^2) / x.

Verify that y = x / (x + C) is a general solution for the given differential equation and show that all solutions contain (0,0) ...?

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  1. 26 September, 16:47
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    We need to show that y = x / (x + c) is a solution of dy/dx = (y - y^2) / x. Then,

    dy/dx = ((x + c) * 1 - x * 1) / (x + c) ^2

    = (x + c - x) / (x + c) ^2

    = c / (x + c) ^2

    and

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

    = (x (x + c) - x^2) / (x (x + c) ^2)

    = (x^2 + cx - x^2) / (x (x + c) ^2)

    = cx / (x (x + c) ^2)

    = c / (x + c) ^2

    which proves the equality.
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