Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Seperation of Variables to find exact solution

  1. Dec 2, 2011 #1
    Can anyone help me please or point me in the right direction, I am needing to find an exact solution for this equation by using seperation of variables and compare them to answers i have calculated for Euler's method & Euler-Cauchy method. The equation is dx/dt=x^2/(t+1) when x(0)=1 and t=time from 0 to 0.5 in 0.1 increments.
    Any pointers would be great thanks.
    Robcru1
     
  2. jcsd
  3. Dec 2, 2011 #2

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    Well, you titled this "Separation of Variables" so apparently you know you can write this as [itex]x^{-2} dx= (t+1)^{-1}dt[/itex] and integrate. If the right side is giving you trouble, let u= t+1.
     
  4. Dec 2, 2011 #3
    Wow quick response.
    Yes i get that, what i am struggling with is where to go from this point and what it is i am actually trying to figure out i have previously rewritten it as ∫1/x^2 dx/dt dt=∫1/t+1 dt
    which got me to ∫1/x^2 dx =∫1/t+1 dt
    = ln(x^2) = ln(t+1)+C
    Then i get stuck & confused
    is this correct?
     
  5. Dec 3, 2011 #4
    Can anyone assist me with this issue, its beginning to bug me now and i seem to be getting nowhere.
    Thanks a lot
     
  6. Dec 3, 2011 #5

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    I'm, sorry, I had kind of thought (apparently foolishly) that, if you are asking about differential equations, you would know how to do basic integrals.

    The integral of [itex]x^n[/itex] is [itex]x^{n+1}/(n+1)+ C[/itex].
     
    Last edited: Dec 3, 2011
  7. Dec 3, 2011 #6
    Yes i do know that.
    I am trying to find the exact solution for x based on the time steps of t from 0 (0.1) 0.5
    are you saying what i have done so far i wrong? Can you point me down the correct path please.
     
  8. Dec 3, 2011 #7

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    Then why did you write that non-sense about the solution being "ln(x^2) = ln(t+1)+C"?

    Also, the solution to a differential equation depends upon continuous change in the independent variable. You cannot have an "exact" solution with "time steps of t from 0 (0.1) 0.5".
     
  9. Dec 3, 2011 #8
    My appologise for coming across as an idiot.
    My original problem was to calculate answers for x^2/(t+1) for time steps 0 (0.1) 0.5 using both eulers method & euler-cauchy method and compare these results against an exact solution using seperation of variables, i presumed i would need to calculate an exact solution at the same time intervals.
    i have done the euler & euler-cauchy calculations and just need to carry out my comparison and tabulate it, this is where i am struggling.
    I am just trying to get some understanding and method behind what i need to do to achieve this.
    If my posts are NON-SENSE then please point me in as to where and which way to progress.
    Many thanks

    Robcru1
     
  10. Dec 4, 2011 #9
    Can anyone help me any further with this please
    My original equation was:- dx/dt=x^2/(t+1)
    What i have so far is:-
    ∫x^-2dx=∫(t+1)^-1dt
    x^-1/-1=∫(t+1)
    Any help is appreciated
     
  11. Dec 5, 2011 #10
    As already pointed out, in general,
    [itex]\int\frac{1}{f(x)}dx \neq \ln{|f(x|})[/itex]

    However, it is true that:
    [itex]\int\frac{1}{t+c}dt=\ln{|t+c|}[/itex] where c is a constant.

    The equation given by HallsofIvy is of course not valid for the special case where n=-1.

    So, after integrating, you get..? The only thing left then is to rewrite the equation into the form x=f(t).
    I suggest (re)reading a calculus book to clear the fog.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Seperation of Variables to find exact solution
Loading...