       Re: Help with solving ODE

• To: mathgroup at smc.vnet.net
• Subject: [mg81796] Re: Help with solving ODE
• From: Jean-Marc Gulliet <jeanmarc.gulliet at gmail.com>
• Date: Wed, 3 Oct 2007 06:14:37 -0400 (EDT)
• Organization: The Open University, Milton Keynes, UK
• References: <fdvdq7\$sul\$1@smc.vnet.net>

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:
>
>
> Further information is also available at sci.physics.research
>
>
> 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. Thanks

With Mathematica, you can use DSolve for an analytic solution. (In the
example below, I have replaced capital 'I' by 'i' since capital I stands
for the imaginary unit, Sqrt[-1], in Mathematica.)

In:= sol = DSolve[{i y''[t] + k y'[t] == 2 G M m d/a^2, y == 0,
y' == 0}, y,
t][]

Out=
(k t)/i      (k t)/i
2 d G m M (i - E        i + E        k t)
{y -> Function[{t}, -----------------------------------------]}
(k t)/i   2  2
E        (a  k )

In:= i y''[t] + k y'[t] == 2 G M m d/a^2 /. sol // Simplify

Out= True

In:= y[t] /. sol /. {i -> 13138117.34, k -> 724.68, G ->
6.67*10^(-8), M -> 158100,
m -> 729.8, d -> 93.09, a -> 22.10}

Out=
-6            7             7  0.0000551586 t
(5.58621 10   (1.31381 10  - 1.31381 10  E               +

0.0000551586 t        0.0000551586 t
724.68 E               t)) / E

In:= % /. t -> 0

Out= 0.

In:= D[%%, t] /. t -> 0

Out= 0.

In:= \$Version

Out= "6.0 for Microsoft Windows (32-bit) (June 19, 2007)"

Regards,
--
Jean-Marc

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