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Math Help - 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:

    \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:

    \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 \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 \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 B(x) somehow depened on f(x) as an input.

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

    Then the equation would be

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

    This is a 2nd order linear ODE
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