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: Solving a system of ODE with multiple 'time' variables

  1. Jan 2, 2012 #1
    1. The problem statement, all variables and given/known data
    Hi everyone,

    Consider the following system of (first order) differential equations:
    [itex]\dot{x}=f(t_1,x,y,z) [/itex]
    [itex]\dot{y}=g(t_2,x,y,z) [/itex]
    [itex]\dot{z}=h(t_3,x,y,z) [/itex]

    where [itex]\dot{x}=\frac{\partial x}{\partial t_1}[/itex], [itex]\dot{y}=\frac{\partial y}{\partial t_2}[/itex], and [itex]\dot{z}=\frac{\partial z}{\partial t_3}[/itex].

    2. Relevant equations

    All existence theorems I know (picard and peano) are formulated such that [itex]t_1=t_2=t_3[/itex], but I'd like to know how to extend these results to the cae shown above.

    3. The attempt at a solution

    I've tried reading the proofs to see if I can figure out a way to apply them to this problem, but I can't see how...Does someone knows whether these theorems hold true when [itex]t_1 \neq t_2 \neq t_3[/itex]? Any help/reference where to look for such theorem would be greatly appreciate!!!
  2. jcsd
  3. Jan 2, 2012 #2
    I'm no ODE guru but it seems to me you need some type of relation between the three t variables.
  4. Jan 2, 2012 #3
    Thanks for your answer. These variables are independent....what I want are conditions of
    the [itex] f [/itex]'s functions to ensure existence and uniqueness without having to solve the system...any idea how I should proceed???

  5. Jan 2, 2012 #4
    Using different differentiation rules, and given a relations [itex]t_2=t_2(t_1)[/itex] and [itex]t_3=t_3(t_1)[/itex], you should be able to reduce the problem to a set of (possibly nonlinear nonautonomous) delay differential equations.

    Sorry, I missed your last comment.
  6. Jan 2, 2012 #5
    No problem...should I infer that without this relation between the t variables I cannot proceed any further???
  7. Jan 3, 2012 #6
    Not neccesarily. You should still be able to solve the system. Just keep in mind that since each f, g and h are implicitly dependent on all the t's, so all of x, y and z will also be dependent an all the t's. Also, you won't get neccesarily get constants as initial contitions, but functions. For example, the initial condition for x might be a function of t2 and t3.
  8. Jan 3, 2012 #7
    ok...understood. As initila conditions I have [itex]f(0,x,y,z)=c_1[/itex], [itex]g(0,x,y,z)=c_2[/itex], and [itex]h(0,x,y,z)=c_1[/itex], where [itex]c[/itex]'s are constants. I don't want to actually solve the system, I want conditions on the f,g, and h functions that ensures existence (and uniqueness) of some solution. Is there any theorem that might help me?
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook