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Math Help - y' = 1 - y^2 Nonlinear 1st order ODE help

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    y' = 1 - y^2 Nonlinear 1st order ODE help

    Hi,
    I'm trying to solve y' = 1 - y^2.

    Using partial fractions I ended up with y = (Ae^2x - 1)/(Ae^2x +1). However, this differs to an answer I was given by someone else. Am I correct?

    Cheers,
    Ben
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  2. #2
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    Quote Originally Posted by weedingb View Post
    Hi,
    I'm trying to solve y' = 1 - y^2.

    Using partial fractions I ended up with y = (Ae^2x - 1)/(Ae^2x +1). However, this differs to an answer I was given by someone else. Am I correct?

    Cheers,
    Ben
    solve dy/dx = 1 - y^2 - Wolfram|Alpha
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  3. #3
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by weedingb View Post
    Hi,
    I'm trying to solve y' = 1 - y^2.

    Using partial fractions I ended up with y = (Ae^2x - 1)/(Ae^2x +1). However, this differs to an answer I was given by someone else. Am I correct?

    Cheers,
    Ben
    I got your answer, and it is equivalent to the answer shown in the link provided by Mr. F. What was the answer you got from someone else? Ah, it doesn't matter, your answer's fine.
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  4. #4
    MHF Contributor chisigma's Avatar
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    Quote Originally Posted by weedingb View Post
    Hi,
    I'm trying to solve y' = 1 - y^2.

    Using partial fractions I ended up with y = (Ae^2x - 1)/(Ae^2x +1). However, this differs to an answer I was given by someone else. Am I correct?

    Cheers,
    Ben
    The first order DE y^{'} = 1-y^{2} is of the type 'separable variables' and its solution seems at first very confortable...

    \displaystyle \frac{dy}{1-y^{2}} = dx (1)

    A subtil pitfall however exists because is...

    \displaystyle \int \frac{dy}{1-y^{2}}=\left\{\begin{array}{ll}\tanh^{-1} y ,\,\,|y|< 1\\{}\\ \coth^{-1} y ,\,\, |y|>1\end{array}\right. (2)

    ... so that the solution is...

    \displaystyle y(x) =\left\{\begin{array}{ll}\tanh (x+c) ,\,\,|y(0)|< 1\\{}\\ \coth (x+c) ,\,\, |y(0)|>1\end{array}\right. (3)

    Kind regards

    \chi \sigma
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