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Math Help - Solve this nonlinear differential equation

  1. #1
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    Solve this nonlinear differential equation

    Hi, I'd like to solve the following differential equation to find y as a function of x (in fact I know what the answer should be due to the context, but I have no idea how to go about actually solving it). Any ideas would be appreciated!

    1 + \left(\frac{\text{d}y}{\text{d}x}\right)^2 = \frac{y^2}{c^2},

    where c is a constant.
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  2. #2
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    Re: Solve this nonlinear differential equation

    Quote Originally Posted by Amanda1990 View Post
    Hi, I'd like to solve the following differential equation to find y as a function of x (in fact I know what the answer should be due to the context, but I have no idea how to go about actually solving it). Any ideas would be appreciated!

    1 + \left(\frac{\text{d}y}{\text{d}x}\right)^2 = \frac{y^2}{c^2},

    where c is a constant.
    \displaystyle \begin{align*} 1 + \left(\frac{dy}{dx}\right)^2 &= \frac{y^2}{c^2} \\ \left(\frac{dy}{dx}\right)^2 &= \frac{y^2}{c^2} - 1 \\ \left(\frac{dy}{dx}\right)^2 &= \frac{y^2 - c^2}{c^2} \\ \frac{dy}{dx} &= \frac{\pm\sqrt{y^2 - c^2}}{|c|} \\ \pm \frac{1}{\sqrt{y^2 - c^2}}\,\frac{dy}{dx} &= \frac{1}{|c|} \end{align*}

    You should now be able to solve this (it's separable).
    Thanks from Amanda1990
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  3. #3
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    Re: Solve this nonlinear differential equation

    Aha - thanks!
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