2nd Order linear ODEs

I'm having trouble solving this problem:

For the equation x^(2)y′′ - xy′ = 0 , find two solutions, show that they are linearly independent and find the general solution. Hint: Try y = x^(r).

I divided the equation by x^(2) first to simplify it, and then I found the first and second derivatives of y and tried plugging them back in. But I don't know where to go from there. Please help! Thank you!

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  1. Hmmm. I'd let v = y'
    Then we have

    x^2 v' = xv
    v' = v/x
    dv/v = dx/x
    log v = log x + c
    v = e^c x
    v = c1 * x
    That means y' = c1 * x
    y = c1 x^2 + c2

    y' = 2c1 x
    y" = 2c1
    x^2 y" - xy' = x^2(2c1) - x(2c1 x) = 0

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    posted by Steve

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