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Re: DSolve difficulties ...

• To: mathgroup at smc.vnet.net
• Subject: [mg111867] Re: DSolve difficulties ...
• From: Slide <wdflannery at aol.com>
• Date: Mon, 16 Aug 2010 05:54:03 -0400 (EDT)
• References: <i45rdj$g3o$1@smc.vnet.net> <i48jgi$e46$1@smc.vnet.net>

On Aug 15, 7:36 am, Helen Read <h... at together.net> wrote:
> On 8/14/2010 6:33 AM, Slide wrote:
>
> > Yesterday I typed in the following command line, into the student
> > version of Mathmatica .
>
> > DSolve[y''[x] == 1/(y[x]*y[x]), y[x], x]
>
> > and got a complicated Solve command and expression back with the
> > comment 'The equations appear to involve the variables to be solved
> > for in an essentially non-algebraic way.'
>
> Adding to my previous post:
>
> OK, the solution to DSolve comes out like this:
>
> Solve[((1/(C[1]^(3/2)))
>      Log[-2 + 2 (C[1] + Sqrt[C[1]] Sqrt[C[1] - 2/y[x]]) y[x]] + (
>      Sqrt[C[1] - 2/y[x]] y[x])/C[1])^2 == (x + C[2])^2, y[x]]
>
> That is, Mathematica is able to solve the differential equation, but the
> solution is given implicitly, rather than solved for y[x]. It got to the
> part where it wants to solve for y[x] in terms of x, can't do it, and
> quits there.
>
> If you then evaluate the output (the Solve[ ] ), you will get the
> message about "essentially non-algebraic" which is Mathematica's way of
> saying it cannot solve the equation explicitly. (It would have to begin
> with, if it could.)
>
> --
> Helen Read
> University of Vermont

Thanks for your response.  I started a new session and now I can get
the solution.  Now, the question is, how did Mathematica arrive at this
solution?  Can I get it to explain, or at least list, the steps it
took?

Note that this differential equation is essentially (with a neg.
constant in front) the world's first, that is, it is the differential
equation for an object falling in a gravitational field in one
dimension.  Newton solved it in two dimensions !