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First order differential equation help

  1. Mar 28, 2006 #1
    i've derived the following differential eqn from a problem i'm working on, and i have tried in vain to solve this if any one can give a direction where i should go our how to attack would be greatly appreciated. the eqn is

    [tex]I\,r= -L\dot{I}+\frac{3}{2}\mu_{0}m R^{2}\frac{z \dot{z}}{\left(R^{2}+z^{2}\right)^{5/2}} [/tex]

    where [tex] r,L,m,\mu_{0},R,\dot{z} [/tex] are all constants. one of two ways i've tried solving this, was since
    [tex]\dot{z}=const.\Rightarrow z=\dot{z}t [/tex]
    which just gives a particular solution i cannot find a solution for. the homogeneous part is quite trivial with the solution being
    [tex] I_{homogeneous}=const. \, \exp(-rt/L)[/tex]
    am i missing something, is there another way. any help please
  2. jcsd
  3. Mar 28, 2006 #2


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    If L = 0, then if r = 0, I can be anything. If r is not zero, then I is just the right side divided by r. So we may assume L is nonzero. Let A = 1.5μ0mR²(z'³L)-1, B = R/z', D = r/L

    I' + DI = At/(B² + t²)5/2

    The solution to the homogenous equation is, as you know IH = Ce-rt/L. So you just need a particular solution. See if this helps at all.
  4. Mar 28, 2006 #3
    thanks for the cite reference, i checked it out, and it's similar information i have in my old ode book in front of me. the integral you get with this
    [tex] \int e^{Dt}\frac{At}{(B^{2}+t^{2})^{5/2}}dt[/tex]
    doesn't seem to be able to be integrated. i think this particular ode might need numerical methods to solve.
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