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 !