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Use u = y' substitution to solve (y + 1) y'' = (y')^2

  1. Mar 11, 2014 #1

    s3a

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    1. The problem statement, all variables and given/known data
    Solve the following differential equation, by using the substitution u = y'.:
    (y + 1) y'' = (y')^2

    2. Relevant equations
    I'm assuming: Chain Rule

    3. The attempt at a solution
    My problem is, simply, that I don't get how to go from u = y' to y'' = u du/dy, and I would appreciate it if someone could show me how!

    Other than that, I should be fine.
     
  2. jcsd
  3. Mar 11, 2014 #2

    maajdl

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    Gold Member

    Re-write:

    (y+1) u' = u²

    then rearrange to get u's on the left and y's on the right and integrate.
    (this is a classical manipulation in classical mechanics)
     
  4. Mar 11, 2014 #3

    LCKurtz

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    Gold Member

    I will use prime for differentiation with respect to the independent variable which I will assume to be ##x##. We have the substitution ##u(y) = y'##. Differentiating both sides with respect to ##x## gives$$
    y'' = u'(y) = \frac{du}{dy}\cdot y' = u \frac{du}{dy}$$It's just the chain rule and I switched the factors in the last step.
     
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