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Re: Help with solving ODE

  • To: mathgroup at smc.vnet.net
  • Subject: [mg81811] Re: [mg81772] Help with solving ODE
  • From: Murray Eisenberg <murray at math.umass.edu>
  • Date: Thu, 4 Oct 2007 04:19:03 -0400 (EDT)
  • Organization: Mathematics & Statistics, Univ. of Mass./Amherst
  • References: <200710030631.CAA28989@smc.vnet.net>
  • Reply-to: murray at math.umass.edu

Did you even TRY to read the documentation about solving differential 
equations in Mathematica?  And learn the most basic things about syntax 
and the function DSolve to solve differential equations?

I'll use i instead of I since the latter is the complex number, and c 
instead of D since the latter stands for derivative, in Mathematica.

You obtain the solution in terms of the constants from:

    soln=First@DSolve[{i y''[t]+k y'[t]==-c/a^2,y[0]==0,y'[0]==0},y[t],t]
{y[t] ->
     -((c*(i - E^((k*t)/i)*i + E^((k*t)/i)*k*t))/(a^2*E^((k*t)/i)*k^2))}

The form of the solution is that of a replacement rule.  If you want 
just the solution itself, as a function of t:

    y[t] /. soln
-((c*(i - E^((k*t)/i)*i + E^((k*t)/i)*k*t))/(a^2*E^((k*t)/i)*k^2))

This is all still in terms of the constants.  To get it with the 
numerical values of the constants, just replace them with their values:

    y[t]/.soln/.{i->13138117.34, k->724.68,
      c->2*6.67*10^-8*158100*729.8*93.09, a->22.10 } // Expand
-73.39232171339471/E^(0.00005515858788942739*t) +
  73.39232171339471*E^(0.*t) - 0.004048216827637412*E^(0.*t)*t

For each input above, I actually applied the function InputForm to the 
corresponding output in order to obtain a one-dimensional format for 
pasting into a plain text e-mail message. In reality, in a Mathematica 
notebook you would not do that so as to allow two-dimensional display 
with fractions and exponents.

Pioneer1 wrote:
> Hi,
> 
> Can anyone help solve this linearized differential equation:
> 
> Iy'' + ky' = 2GMmd/a^2
> 
> Primes are time derivates of y (=theta=excursion angle). Is it
> possible to solve this for the initial conditions y(0)=0 and y'(0)=0?
> 
> I got the solution at sci.math for the non-linear version and I want
> to compare the two. Here's the link to sci.math thread:
> 
> http://groups.google.com/group/sci.math/browse_thread/thread/a6ee2f782df09625/53cf5573d354a3ab#53cf5573d354a3ab
> 
> Further information is also available at sci.physics.research
> 
> http://groups.google.com/group/sci.physics.research/browse_thread/thread/d391940cc173f9dc/eed90e6c3fee0edc#eed90e6c3fee0edc
> 
> Parameters are:
> 
>> y = theta = excursion angle in radians
>> A = I = moment of inertia = 13,138,117.34 g cm^2
>> B = R = damping = for now I assume this to be zero
>> C = k = torsion constant  = 724.68 g cm^2 sec^-2
>> d = moment arm = 93.09 cm
>> D = 2GMmd = 2 * 6.67*10^-8 * 158100 * 729.8 * 93.09 = 1432.82
>> a = distance between weights = 22.10 cm
> 
> I would truly appreciate help with this. Thanksi
> 
> 

-- 
Murray Eisenberg                     murray at math.umass.edu
Mathematics & Statistics Dept.
Lederle Graduate Research Tower      phone 413 549-1020 (H)
University of Massachusetts                413 545-2859 (W)
710 North Pleasant Street            fax   413 545-1801
Amherst, MA 01003-9305


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