Re: Solve fails

• To: mathgroup at smc.vnet.net
• Subject: [mg33346] Re: [mg33326] Solve fails
• From: Andrzej Kozlowski <andrzej at lineone.net>
• Date: Sat, 16 Mar 2002 01:40:35 -0500 (EST)
• Sender: owner-wri-mathgroup at wolfram.com

```Solve is not meant to find approximate solutions at all, and is not
really  intended to be used to solve non-algebraic ones. It does of
course "solve" some simple non-algebraic equations in terms of "Inverse
Functions", e,g. it will give some solutions to Solve[Sin[x^3] == 1, x].
So, the issue being raised, is, I assume, whether Solve ought to find
the "trivial" solutions, like x==0,x=1, x=-1 for it clearly cannot be
expected to find x=+-2.0253909752730426 (this is , a job for FindRoot).
These "trivial" solutions are "essentially algebraic" since they just
the solutions x^3==x. In some similar cases Mathematica does get the

In[1]:=
Solve[ArcSin[x] == ArcSin[x^3], x]

Out[1]=
{{x -> -1}, {x -> 0}, {x -> 1}}

In[2]:=
Solve[Exp[x] == Exp[x^3], x]

From In[2]:=
Solve::ifun:Inverse functions are being used by Solve, so some solutions
may \
not be found.

Out[2]=
{{x -> -1}, {x -> 0}, {x -> 1}}

In[3]:=
Solve[Log[x] == Log[x^3], x]

Out[3]=
{{x -> -1}, {x -> 1}}

but it others it does not. Should it? Well, I do not know, and I do not
think it is terribly important, since all such solutions lie clearly
"beyond the call of duty" of Solve. However, evaluating

Trace[Solve[x*Sin[x] == x*Sin[x^3], x], TraceInternal -> True]

seems to show that Solve does notice that the equation is equivalent to
x==0 || Sin[x]-Sin[x^3]==0, so it ought at least to give 0 as a solution
but then it somehow manages to  "loose" it and returns {{}}.

Andrzej Kozlowski
Toyama International University
JAPAN
http://platon.c.u-tokyo.ac.jp/andrzej/

On Friday, March 15, 2002, at 12:51  AM, Richard Fateman wrote:

>
> Solve[x*Sin[x]==x*Sin[x^3],x] gives {{}}
>
> but
> Solve[Sin[x]==Sin[x^3],x]  returns unchanged
>
> each gives a warning.
>
> There are very many solutions, among them x=0, x=+-1,
> x=+-2.0253909752730426 approximately
>
>
>
>

```

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