1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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
    $$
    y'\arctan(x)-\frac{y}{1+x^2}=0\\
    \Rightarrow \int \frac{\mathrm{d}y}{y}=\int \frac{1}{1+x^2}\frac{1}{\arctan(x)}\, \mathrm{d}x\text{.}
    $$
    At this point it is helpful to note that
    $$
    \frac{\mathrm{d}}{\mathrm{d}x}\arctan(x)=\frac{1}{1+x^2}\text{.}
    $$
    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.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Separation of Variables for ODE
  1. Separating an ODE (Replies: 3)

Loading...