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Thread: power series solutions

  1. #1
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    power series solutions

    $\displaystyle xy'=y$
    I have to find a power series solution to this equation of the form $\displaystyle \sum a_{n}x^{n}$
    Then solve the equation directly.

    I have worked out by studying the equation that the solution is $\displaystyle nx$ but i can't work out how to obtain that answer.

    My working so far is:
    $\displaystyle xy'-y=0$
    $\displaystyle y'-\frac{y}{x}=0$
    $\displaystyle \sum na_{n}x^{n-1} - \frac{1}{x}\sum a_{n}x^{n}=0$
    (both of sums go from $\displaystyle n=0$ to $\displaystyle \infty$ I wasn't sure how to write those in LaTex.)

    Thanks in advance
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  2. #2
    Behold, the power of SARDINES!
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    I solve the wrong equation
    Last edited by TheEmptySet; Jun 12th 2009 at 08:20 AM. Reason: I solved the wrong equation. Thanks Moo
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  3. #3
    Moo
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    Hello,

    For this kind of problems, you should write the starting indice
    It's important.

    Let $\displaystyle y=\sum_{n\geq 0} a_nx^n$ be the solution of the equation.

    $\displaystyle y=a_0+\sum_{n\geq 1}a_nx^n$. Then $\displaystyle y'=\sum_{n\geq 1}na_nx^{n-1}$

    $\displaystyle \begin{aligned}xy'=y &\Longrightarrow x\sum_{n\geq 1} na_nx^{n-1}=\sum_{n\geq 0} a_nx^n \\
    &\Longrightarrow \sum_{n\geq 1} na_nx^n=a_0+\sum_{n\geq 1} a_nx^n \\
    &\Longrightarrow a_0+\sum_{n\geq 1} \{a_nx^n-na_nx^n\}=0 \end{aligned}$

    Let $\displaystyle x=0$ to get $\displaystyle a_0=0$

    Differentiate successively, and let $\displaystyle x=0$ to get $\displaystyle a_n-na_n=0 \quad \quad \forall n\geq 1$

    Thus $\displaystyle a_n(n-1)=0 \quad \forall n\geq 1$

    So either n=1, either $\displaystyle a_n=0$

    So all the possible solutions are :
    - $\displaystyle a_n=0 \quad \forall n\geq 0$ ---> $\displaystyle y=0$ is a solution

    - $\displaystyle a_n=0 \quad \forall n\geq 2$ and $\displaystyle a_0=0$. And $\displaystyle a_1$ can be whatever you want. So $\displaystyle y=kx$ is a solution.


    @ Tessy : you got the wrong equation
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  4. #4
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    Thanks Moo I solve the wrong equation
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