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Unit problem in differential equation

  1. Jul 20, 2011 #1
    I have one differential equation like
    dN/dt=c*(other terms with no unit)
    unit of c is 1/ns. Now if I solve this equation, I will get value of N corresponding to t.
    (1) Then what will be the unit of t?
    (2) and if I calculate dN/dt1=(other terms with no unit). where t1=t/c.
    Then what will be the unit of t1?
     
  2. jcsd
  3. Jul 20, 2011 #2

    hunt_mat

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    As I see it you have three possible units [N], [t] and [c], and I nonlimensionalise according to [itex]N=N_{d}\bar{N}[/itex], [itex]t=t_{d}\bar{t}[/itex] and [itex]c=c_{d}\bar{c}[/itex]. There I put all the dimensions into the quantities with subscript d. So this turns the differential equation into:
    [tex]
    \frac{d\bar{N}}{d\bar{t}}=\frac{c_{d}t_{d}}{N_{d}}\bar{c}
    [/tex]
    Where [itex]c_{d}t_{d}/N_{d}[/itex] is a nondimensional quantity. Does this clear things up?
     
  4. Jul 20, 2011 #3
    t will be in ns. Your RHS has dimensions of time-1. dN/dt also has dimensions of time-1. So everything matches.

    Are you sure you want to divide t by c? Maybe I'm confused, but I think it would make more sense to multiply t by c, in order to get a dimensionless version of the equation. For instance, if you define [itex]\tau=ct[/itex], then you get [itex]\frac{dN}{d\tau}=\mbox{(other terms)}[/itex] and c has gone away.
     
  5. Jul 20, 2011 #4
    --------------------------
    Thank you. Sorry for mistake. I have to multiply t by c. Then I will get value of N for different times but then time has no unit.
     
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