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Find the differential equation or system of differential equations

  1. Feb 3, 2014 #1
    Find the differential equation or system of differential equations ***

    Find the differential equation or system of differential equations assoicated with the following flows
    a) ##\phi_t (x) = \frac{x}{\sqrt{1-2x^2t}} ## on ##{\mathbb R} ##

    b) ##\phi_t (x,y) = (xe^t, \frac{y}{1-y^t}) ## on ##{\mathbb R}^2 ##

    The ways I solve these two questions are that I simply take the derivatives of them

    for (a), ##\left.\dfrac{d}{dt}\right|_{t=0} \phi_t (x)## if this is the right way, check you check my answer, ##\frac{-x}{2} - 2x^2##

    for (b), ##\left.\dfrac{d}{dt}\right|_{t=0} \phi_t (x,y)## if this is the right way, check you check my answer, ##(xe^t, \frac{ty}{(1-y^t)^2})##
     
    Last edited: Feb 3, 2014
  2. jcsd
  3. Feb 3, 2014 #2

    haruspex

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    In (a), I assume that what you are looking for is a differential equation involving only ϕ, dϕ/dx and x.
    If so, you need to differentiate wrt x, and you can't get rid of the t just by evaluating at t=0.
     
  4. Feb 3, 2014 #3
    Just edit my question...please take a look again to see if anything changes to your response
     
  5. Feb 4, 2014 #4

    haruspex

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    Yes, it means I'm a bit out of my depth.. but I think you need to bear in mind that x = x(t), and it's dϕ/dt, not ∂ϕ/∂t.
     
  6. Feb 4, 2014 #5

    Dick

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    I'm not an expert in the subject either, but in flow problems like this I think you have ##x(t)=\phi(x,t)##. Treat the x in ##\phi(x,t)## as a constant. And I'd be interested in how Askhwhelp got either solution. I think the first one is just plain wrong. And for another thing, they don't look like differential equations to me and the second one even has a t in it. How can that be if you set t=0?
     
    Last edited: Feb 4, 2014
  7. Feb 4, 2014 #6

    haruspex

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    Having read up on this, I think I understand it well enough now to give a more helpful answer.
    As Dick says, it should be partial differentiation, the result should be equated to dx/dt, and that answer is wrong.
    Please post your working.
     
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