1. Limited time only! Sign up for a free 30min personal 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!

Homework Help: Theorem of existence and unicity.

  1. Mar 3, 2012 #1
    1. The problem statement, all variables and given/known data
    In the following case: x'(t) = log (3t (x(t) - 2)) does the theorem of existence and unicity guarantee a unique solution for the initial value problem x(3) = 5, justify your answer?

    2. Relevant equations
    x'(t) = log (3t (x(t) - 2))

    3. The attempt at a solution
    Ok what I would do is take the domain of f(x,t) and the domain of ∂f(x,t)/∂x and then find the common interval and if the initial value problem is within this interval then there is a unique solution. However there is x(t) and t in the function so I don't understand how to find the domain?
    Last edited: Mar 3, 2012
  2. jcsd
  3. Mar 3, 2012 #2


    User Avatar
    Science Advisor

    For what x and t is that function, log(3t(x- 2), continuous and differentiable? What is the largest interval containing (3, 5) on which the function is continuous and differentiable?
  4. Mar 3, 2012 #3
    Yeh, how would I find that out? Thats what i'm confused about.

    EDIT: Would I just plug numbers into the x and t? Until I got an x and t that came up with 'Math error'? Because if thats the case then when t = 0 the function is discontinuous and when x = 2? Am I right?
    and then the partial derivative is 3t/(3t(x-2)) = 1/(x-2) in which case x cannot equal 2 and t is eqal to all real numbers? So the common intervals are x cannot = 2 and t cannot = 0, with intial values t_0 = 3 and x_0 = 5 which are within the common intervals, therefore there is a unique solution? On a side note, should i have taken the partial derivative with respect to t or x? Or do I take partial derivative with respect to both and then compare all 3 intervals.
    Last edited: Mar 3, 2012
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook