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Math Help - first order non-linear differential equation

  1. #1
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    first order non-linear differential equation

    I am trying to solve this differential equation and I can't get started and was wondering if I could get some help with a kick start:

    (\frac{dy}{dx})^2 +(\frac{dy}{dx})*x-\frac{dy}{dx}-x=0

    Any help would be greatly appreciated!

    MT
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  2. #2
    A Plied Mathematician
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    I know it sounds crazy, but couldn't you use the quadratic formula here to solve for y'? That is, you have

    (y')^{2}+y'(x-1)-x=0, so

    y'=\dfrac{-(x-1)\pm\sqrt{(x-1)^{2}-4(-x)}}{2}\dots

    Does that work?
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  3. #3
    Behold, the power of SARDINES!
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    Quote Originally Posted by Empty View Post
    I am trying to solve this differential equation and I can't get started and was wondering if I could get some help with a kick start:

    (\frac{dy}{dx})^2 +(\frac{dy}{dx})*x-\frac{dy}{dx}-x=0

    Any help would be greatly appreciated!

    MT
    Notice that

    \displaystyle \left(\frac{dy}{dx}\right)^2 +\left(\frac{dy}{dx}\right)x-\frac{dy}{dx}-x=0 \iff \left(\frac{dy}{dx}\right)^2 -\frac{dy}{dx}+\left(\frac{dy}{dx}\right)x-x=0

    factoring gives

    \displaystyle \frac{dy}{dx}\left( \frac{dy}{dx}-1\right)+x\left( \frac{dy}{dx}-1\right)=0

    \displaystyle \left( \frac{dy}{dx}-1\right)\left( \frac{dy}{dx}+x\right)=0

    Use the zero product principle. This will give two different solutions. Remember for nonlinear equations that the principle of superposition does not hold. So you cannot add the solutions to obtain a more general solution.
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  4. #4
    A Plied Mathematician
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    Reply to TheEmptySet:

    Ah, your method is more elegant, because you don't get those annoying magnitude signs as follows:

    y'=\dfrac{-x+1\pm|x+1|}{2}.
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  5. #5
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    Thank you all for your help... I should have known to use the quadratic equation!! Thanks TheEmptySet for that alternative approach!
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