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I 2nd order non-linear ODE

  1. Feb 1, 2017 #1

    joshmccraney

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    Can someone check my work here? Both ##f=f(x)## and ##y=y(x)##.
    $$f'y'+\frac{fy''}{1+y'^2}=0\implies\\
    \frac{y''}{y'(1+y'^2)}=-\frac{f'}{f}\\
    \frac{y''}{y'(1+y'^2)}=-\ln(f)$$
    Now let ##v=y'##, which implies
    $$
    \int\frac{1}{v(1+v^2)}\,dv=-\int\ln(f)\,dx\implies\\
    \ln(v) - \frac{1}{2}\ln(1 + v^2)+C=-\int\ln(f)\,dx\implies\\
    \frac{y'}{\sqrt{1 + y'^2}}=k\exp\left[-\int\ln(f)\,dx\right]$$
    Am I correct to this point? Also, how would you proceed?
     
  2. jcsd
  3. Feb 1, 2017 #2
    This last line is not correct. The right side should be [itex] -\frac{d}{dx} \ln f\left(x\right) [/itex]
     
  4. Feb 2, 2017 #3

    joshmccraney

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    Gold Member

    Thanks!!!
     
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