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[MUG] Re: dsolve does not find second solution
| [MUG] Re: dsolve does not find second solution |
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Author: Maple User Group
Posted: Tue, 17 Dec 2002 13:07:28 -0500
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>> From: Maple User Group "maple_gr"
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Date: Mon, 16 Dec 2002 08:23:00 -0500 (EST)
From: Stephen Forrest "saforres"
To: "maple-list"
Subject: dsolve does not find second solution
On Mon, 16 Dec 2002, Helmut Kahovec wrote:
[snip]
> 2 2
> 3/2 (x + 1) _C1 - 1/2 sqrt((x + 1) _C1 + 4)
> sol1 := y(x) = 1 - ---------------------------------------------
> _C1
>
> There should be a second solution, which is missing:
>
> > sol2:=y(x)=1-(3/2*(x+1)*_C1+1/2*((x+1)^2*_C1^2+4)^(1/2))/_C1;
>
> 2 2
> 3/2 (x + 1) _C1 + 1/2 sqrt((x + 1) _C1 + 4)
> sol2 := y(x) = 1 - ---------------------------------------------
> _C1
Observe that sol2 is equal to sol1 after the substitution _C1 -> -_C1,
thus since _C1 can be any real, sol2 is contained in dsolve's solution.
Steve
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From: "Edgardo S. Cheb-Terrab" "ecterrab"
To: "maple-expert" "maple-list"
Subject: [maple-expert] dsolve does not find second solution
Date: Mon, 16 Dec 2002 16:01:17 -0800
Hi,
I see no solutions missing in dsolve's output - no bug.
Given two general solutions S1(y, x, _C1) and S2(y, x, _C1), we say "these
solutions are the same" (so there is only one) whenever
S2(y, x, _C1) = S1(y,x, F(_C1) )
for some F. In simpler words: whenever S2 can be obtained from S1 by
redefining the arbitrary constant _C1~ = F(_C1), then S2 is not a solution
different than S1.
For example, I can always change _C1 -> F(_C1), in the solution sol1 you show,
with whatever F of my choice, and obtain as many different forms of the
solution as I want - they are all"the same solution.
In particular, if I change _C1 = -_C1 in the sol1 you show I obtain your sol2,
from where sol2 is not another solution and therefore it is not the case that
dsolve is missing one solution.
Edgardo
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