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Thread: Interesting 2nd order non-linear DE

  1. #1
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    Interesting 2nd order non-linear DE

    Hello,

    I have recently come across an interesting second order non-linear homogeneous DE whose form I cannot seem to find a general solution to. It loos like such:

    $\displaystyle \dfrac{d^2}{dx^2}f(x) + B(x)*\dfrac{d}{dx}f(x) + f(x)*\dfrac{d}{dx}B(x) = 0$

    Where B(x) is a arbitrary function. Has anyone seen this before or have any hints?

    Thanks
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  2. #2
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    Quote Originally Posted by KrayzBlu View Post
    Hello,

    I have recently come across an interesting second order non-linear homogeneous DE whose form I cannot seem to find a general solution to. It loos like such:

    $\displaystyle \dfrac{d^2}{dx^2}f(x) + B(x)*\dfrac{d}{dx}f(x) + f(x)*\dfrac{d}{dx}B(x) = 0$

    Where B(x) is a arbitrary function. Has anyone seen this before or have any hints?

    Thanks
    Notice that

    $\displaystyle \displaystyle \frac{d}{dx}\left[B(x)f(x)\right]=B(x)\frac{df}{dx}+f(x)\frac{dB}{dx}$

    This gives the equation as

    $\displaystyle \displaystyle \frac{d}{dx}\left[ \frac{df}{dx}+B(x)f(x)\right]=0$

    Now just integrate both sides and you have a first order equation that can be solved via an integrating factor.
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  3. #3
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    It's nonlinear?
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  4. #4
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    Thanks

    You're right! I'm embarrassed that I missed that .

    Thank you, sir.

    (Although interestingly, alpha couldn't solve it either f''(x) + b(x)*f'(x) + b'(x)*f(x) = 0 - Wolfram|Alpha)

    As to Ackbeet, from what I understand, and from what alpha seems to confirm, if B(x) could be non-linear, then the equation is non-linear. Please correct me if I'm wrong! (I could never quite get this straight in school)
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  5. #5
    Behold, the power of SARDINES!
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    Quote Originally Posted by KrayzBlu View Post
    You're right! I'm embarrassed that I missed that .

    Thank you, sir.

    (Although interestingly, alpha couldn't solve it either f''(x) + b(x)*f'(x) + b'(x)*f(x) = 0 - Wolfram|Alpha)

    As to Ackbeet, from what I understand, and from what alpha seems to confirm, if B(x) could be non-linear, then the equation is non-linear. Please correct me if I'm wrong! (I could never quite get this straight in school)
    I would not classify it as nonlinear. The equation just has non constant coefficients. It would only be nonlinear if$\displaystyle B(x)$ somehow depened on $\displaystyle f(x)$ as an input.

    For example is $\displaystyle B(x)=\sin(x) \implies B'(x)=\cos(x)$

    Then the equation would be

    $\displaystyle f''(x)+\sin(x)f'(x)+cos(x)f(x)=0$

    This is a 2nd order linear ODE
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