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Thread: Rewrite DE as linear equation

  1. #1
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    Rewrite DE as linear equation

    Hi,

    Is it possible to rewrite the equations

    (a) $\displaystyle x'=\begin{cases}
    \frac{x^{2}-1}{x-1} & x\neq1\\
    2 & x=1\end{cases}
    $

    (b) $\displaystyle x'=\begin{cases}
    \frac{x^{4}-1}{x^2-1} & x\neq1\\
    2 & x=1\end{cases}
    $
    as linear equations?


    Can someone give me a hint, please?

    Thanks a lot in advance.

    Honey$\displaystyle \pi$
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  2. #2
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    Quote Originally Posted by HoneyPi View Post
    Hi,

    Is it possible to rewrite the equations

    (a) $\displaystyle x'=\begin{cases}
    \frac{x^{2}-1}{x-1} & x\neq1\\
    2 & x=1\end{cases}
    $

    (b) $\displaystyle x'=\begin{cases}
    \frac{x^{4}-1}{x^2-1} & x\neq1\\
    2 & x=1\end{cases}
    $
    as linear equations?


    Can someone give me a hint, please?

    Thanks a lot in advance.

    Honey$\displaystyle \pi$
    I'm hoping you can see that

    $\displaystyle \frac{x^2 - 1}{x - 1} = \frac{(x + 1)(x - 1)}{x - 1}$

    $\displaystyle = x + 1$.

    As $\displaystyle x \to 1, f(x) \to 2$.

    So you can rewrite this function as

    $\displaystyle f(x) = x + 1$ for all $\displaystyle x$.



    You should also be able to see that

    $\displaystyle \frac{x^4 - 1}{x^2 - 1} = \frac{(x^2 + 1)(x^2 - 1)}{x^2 - 1}$

    $\displaystyle = x^2 + 1$.

    As $\displaystyle x \to 1, f(x) \to 2$.

    So you can rewrite the function as

    $\displaystyle f(x) = x^2 + 1$ for all $\displaystyle x$.


    So, you can rewrite the first as a linear function, and the second as a quadratic function.
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