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Math Help - Writing a differential equation in power series

  1. #1
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    Writing a differential equation in power series

    Given the equation:

    (x^2)y''+xy'-y=0

    How do you write this in power series?
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  2. #2
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    Quote Originally Posted by Toucan View Post
    Given the equation:

    (x^2)y''+xy'-y=0

    How do you write this in power series?
    Assume y = a + bx + cx^2 + dx^3 + \dots.


    Then y' = b + 2cx + 3dx^2 + 4ex^3 + \dots

    and y'' = 2c + 6dx + 12ex^2 + 20fx^3 + \dots.


    Substitute this all into the equation...
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  3. #3
    MHF Contributor chisigma's Avatar
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    The DE equation ...

    x^{2}\cdot y^{''} + x\cdot y^{'} - y=0 (1)

    ... belongs to a family which is called 'Euler's equation' and its particular solutions are of the form x^{\alpha}. Substituting in (1) y= x^{\alpha} we obtain that it must be...

    \alpha = \pm 1 (2)

    ... so that ther general solution of (1) is...

    y= c_{1}\cdot x + \frac{c_{2}}{x} (3)

    ... which is itself a Laurent series expansion...

    Kind regards

    \chi \sigma
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