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A seemingly difficult differential equation

  1. Jan 29, 2006 #1
    Anyone know how to solve it?
    Either step by step or using MatLab/Mathematica/Maple is ok.
    [tex]\frac{a+bsinwt-c\sqrt{H}}{k}=\frac{dH}{dt}[/tex]

    I need it in modelling an engineering problem, but I simply don't have any idea to solve it...
     
  2. jcsd
  3. Jan 29, 2006 #2

    benorin

    User Avatar
    Homework Helper

    the solution is approaching ulgy

    Maple and I can only solve it if b=0.

    It is easy for a=b=0, e.g. the homogeneous case, namely

    [tex]k\frac{dH}{dt}+c\sqrt{H(t)}=0[/tex]

    has the solution

    [tex]H(t)=(C-\frac{ct}{2k})^2[/tex].

    If [tex]a\neq 0,[/tex] then the solution is approaching ulgy, see for yourself: Implicitly it is given by

    [tex]c^2t+ka\ln(-a^2+c^2H(t))+2ck\sqrt{H(t)}-ka\ln(a+c\sqrt{H(t)})+ka\ln(c\sqrt{H(t)}-a)+C = 0[/tex]

    have a nice day
     
    Last edited: Jan 29, 2006
  4. Jan 29, 2006 #3
    thanks
    but b cannot be zero.
    anyone have more?
     
  5. Dec 17, 2009 #4
    I think that a solution is as follows:

    H = ( a/c + (H0^2-a/c)exp(-c/2k t) + b c sin(wt)/(c^2+4*k^2*w^2) - 2 k b w cos(wt)/(c^2+4*k^2*w^2) )^2
     
    Last edited: Dec 17, 2009
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