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Re: Mathematica issue

  • To: mathgroup at smc.vnet.net
  • Subject: [mg127861] Re: Mathematica issue
  • From: "Dr. Wolfgang Hintze" <weh at snafu.de>
  • Date: Tue, 28 Aug 2012 04:50:22 -0400 (EDT)
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  • References: <k1epvg$rnq$1@smc.vnet.net>

On 27 Aug., 05:36, Matthias Bode <lvs... at hotmail.com> wrote:
> Hola:
> As neither Solve[] nor Reduce[] find an analytical solution there isn't any - most probably.
> One might, however, do - inter alia - this:
> Clear[a0, a1, b0, b1, c0, d0, eq01, sol01]eq01 = Expand[a0 + b0 +   x^a1*(c0 - d0*x^b1) == 0]; sol01 = FindInstance[eq=
01, {a0, b0, c0, d0,        a1, b1, x}, Reals]a0 + b0 + x^a1*(c0 - d0*x^b1) == 0 /. sol01N[sol01]
> Out[3]= {{a0 -> (46314691670528 - 1684956133536*           134^(7/10) - 15*134^(4/5))/         57893364588160, b0 -> -(4/5),      c0 -> 39/10, d0 -> -(3/2),      a1 -> -(3/10), b1 -> -(49/10), x -> 134}}
> Out[4]= {True}
> Out[5]= {{a0 -> -0.09729070208467958, b0 -> -0.8,      c0 -> 3.9,d0 -> -1.5, a1 -> -0.3,      b1 -> -4.9, x -> 134.}}
> Best regards,
> MATTHIAS BODE
> S 17.35775=B0, W 066.14577=B0
> 2'740 m
> AMSL.
>
>
>
>
>
> > From: n... at 12000.org
> > Subject: Re: Mathematica issue
> > To: mathgr... at smc.vnet.net
> > Date: Sun, 26 Aug 2012 05:45:30 -0400
>
> > On 8/26/2012 3:18 AM, Nasser M. Abbasi wrote:
>
> > >> I need analytical expression for the x in terms of A,B,C,D and a,b,
>
> > > try this:
>
> > > parms = {A0 -> 1, B0 -> 2, a -> 3, C0 -> 4, D0 -> 5, b -> 6}
> > > eq    = A0 + B0 + x^a (C0 - D0 x^b) == 0
> > > sol   = Solve[eq /. parms, x]
>
> > Opps, just noticed you want symbolic solution.
> > Mathematica 8.04 does not do it. I doubt this can be solved symbolicall=
y.
> > Need to use some numbers for the parameters.
>
> > Clear[A0, B0, a , C0, D0 , b]
> > eq = A0 + B0 + x^a (C0 - D0 x^b) == 0
> > Solve[eq, x]
>
> > Solve::nsmet: This system cannot be solved with the methods available t=
o =
> Solve. >>
>
> > --Nasser

If you restrict yourself to integer exponents, the solubility of the
equation is governed by the Abel=96Ruffini theorem:

In algebra, the Abel=96Ruffini theorem (also known as Abel's
impossibility theorem) states that there is no general algebraic
solution=97that is, solution in radicals=97 to polynomial equations of
degree five or higher.

You might want to watch this in Mathematica usind this command

Table[{n, eq = x^n + a*x + b == 0; Simplify[Solve[eq, x]]}, {n, 1, 5}]

More generally you are facing a transcendental quation which has no
closed solution in most cases but must be soved numerically.
Here's an example:

In[57]:= eq1 = x^(-E) + x^Pi == 3

Out[57]= x^(-E) + x^Pi == 3

Watch the behaviour of the function

In[66]:= Plot[{3, x^(-E) + x^Pi}, {x, 0, 2}]
[... the graph of the function]

and then find the solutions (roots) numerically

In[61]:= FindRoot[eq1, {x, 1}]

Out[61]= {x -> 1.34813204709293}

In[62]:= FindRoot[eq1, {x, 0.5}]

Out[62]= {x -> 0.6958252736289695}

Regards,
Wolfgang



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