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Re: yg = \frac{{d(yv)}}{{dt}}, how to solve this differential equation.

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• Subject: [mg125525] Re: yg = \frac{{d(yv)}}{{dt}}, how to solve this differential equation.
• From: Murray Eisenberg <murray at math.umass.edu>
• Date: Sat, 17 Mar 2012 02:52:11 -0500 (EST)
• Delivered-to: l-mathgroup@mail-archive0.wolfram.com

You presented the differential equation using LaTeX syntax, not
Mathematica syntax. This suggest you know essentially nothing about
Mathematica.

To begin: since multi-character names are allowed in Mathematica, you
have to indicate multiplication explicitly, perhaps with just a space,
rather than juxtaposition. Thus:

y g

Second, the Mathematica notation for a function y of a variable t is y[t].

Third, one notation for taking the derivative of a function y of t is
just y'[t]. Another is D[y[t], t], and the latter is more convenient for
taking the derivative of a product such as that of y v:

D[y[t] v[t], t]

Now of course velocity is the derivative of position, so you really have
there:

D[y[t] y'[t], t]

You can either let Mathematica figure out what that is or use the
Product Rule from calculus:

D[y[t] y'[t], t] == (y'[t])^2 + y[t] y''[t]
True

Note the double-equal sign == for indicating an equation.

Finally, use the Mathematica function DSolve to solve a differential
equation. In your example, this will be:

DSolve[g y[t] == D[y[t] y'[t], t], y[t], t]

You probably won't like the pair of solutions you obtain, as they will
be expressed as inverse functions of some rather complicated expressions
involving complex cube- and sixth-rots of -1 along with elliptic integrals.

You may have better luck with tractable solutions if you specify initial
conditions, but I doubt it. So you may have to try for numerical
solutions, use DSolve.

On 3/16/12 7:30 AM, Hongyi Zhao wrote:
> Hi all,
>
> I've a differential equation looks like following:
>
> yg = \frac{{d(yv)}}{{dt}}
>
> where, g is gravity acceleration, y is the displacement, and the v is
> velocity.  Could you please give me some hints by using mathematica to
> solve it?
>
> Best regards

--
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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