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Find solutions of a differential equation of a given form

  1. Let f(x) be a function such that

        \[ 2f'(x) = f \left( \frac{1}{x} \right) \qquad \text{for } x>0, \quad \text{and} \quad f(1) = 2. \]

    Let y = f(x) and show that y satisfies

        \[ x^2 y'' + axy' + by = 0, \]

    for constants a,b. Determine the values of the constants.

  2. Find a solution f(x) = Cx^n.

Incomplete.

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