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Homework Help: Separation of Variables for ODE

  1. Apr 13, 2012 #1
    1. The problem statement, all variables and given/known data

    Solve the following equation by separation of the variables:

    y' tan-1x - y (1+x2)-1 = 0

    2. Relevant equations

    3. The attempt at a solution

    I am not sure if tan-1x stands for arctan x or (tan x)-1. (This has been taken out a book.) Any help on this would be appreciated.

    If arctan x, then we have to integrate 1 / [(1+x2)(arctan x)] w.r.t x. No idea how to do this.

    If (tan x)-1, then we have to integrate tan x / (1+x2) w.r.t. x. No idea on how to do this either.
  2. jcsd
  3. Apr 13, 2012 #2
    It's should be arctan(x). So
    \Rightarrow \int \frac{\mathrm{d}y}{y}=\int \frac{1}{1+x^2}\frac{1}{\arctan(x)}\, \mathrm{d}x\text{.}
    At this point it is helpful to note that
    So, your integral will just be a ##u## substitution. I believe you can take it from here?
  4. Apr 13, 2012 #3
    Yeah, I think I can do the rest. Sub u = arctan(x) and find the answer to be y = k arctan (x).
  5. Apr 13, 2012 #4
    I believe that's correct. And you can always plug your function for y back into the differential equation and make sure that it gives you a true statement.
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