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Re: DSolve for a real function

• To: mathgroup at smc.vnet.net
• Subject: [mg127939] Re: DSolve for a real function
• From: Andreas Talmon l'Armée at smc.vnet.net
• Date: Tue, 4 Sep 2012 05:45:06 -0400 (EDT)
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• Delivered-to: l-mathgroup@wolfram.com
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• References: <k1k8np$90f$1@smc.vnet.net> <20120901062746.B3D7F687B@smc.vnet.net>

Hi,

My initial conditions are the following and I am pretty sure that my 
solution consists only of a real part.
The Solution has four eigenvalues and they are complex conjugated. With 
all variables and all parameters real numbers I must be able to retrieve 
a real solution. But being a mathematica newbie,  I do not understand 
how to do it with mathematca.

y''[-c] == ic0, y''[c] == ic0, y'''[-c] == ic1, y'''[c] == -ic1

My Notebook is also ready for download at dropbox.com:
http://dl.dropbox.com/u/4920002/DGL_4th_Order.nb

Clear[sol]

$Assumptions = {a \[Element] Reals, ic0 \[Element] Reals,
    ic1 \[Element] Reals, c \[Element] Reals};

sol[a_, x_] =
  y[x] /. DSolve[{y''''[x] + a y[x] == 0, y''[-c] == ic0,
       y''[c] == ic0, y'''[-c] == ic1, y'''[c] == -ic1}, y[x],
      x][[1]] // FullSimplify

Reduce[Element[sol[a, x], Reals], a, Reals]



Thanks for your help,

Andreas






On 09/01/2012 08:27 AM, Bob Hanlon wrote:
> What are your initial conditions?
>
> Clear[sol]
>
> sol[a_, x_] = y[x] /. DSolve[
>       {y''''[x] + a y[x] == 0,
>        y[0] == ic0, y'[0] == ic1,
>        y''[0] == ic2, y'''[0] == ic3},
>       y[x], x][[1]] // FullSimplify
>
> (1/(2*a^(3/4)))*
>     (Cosh[(a^(1/4)*x)/Sqrt[2]]*
>          (2*a^(3/4)*ic0*Cos[(a^(1/4)*x)/
>                   Sqrt[2]] + Sqrt[2]*
>               (Sqrt[a]*ic1 + ic3)*
>               Sin[(a^(1/4)*x)/Sqrt[2]]) +
>        (Sqrt[2]*(Sqrt[a]*ic1 - ic3)*
>               Cos[(a^(1/4)*x)/Sqrt[2]] +
>             2*a^(1/4)*ic2*Sin[(a^(1/4)*x)/
>                   Sqrt[2]])*
>          Sinh[(a^(1/4)*x)/Sqrt[2]])
>
> Reduce[Element[sol[a, x], Reals], a, Reals]
>
> a > 0
>
>
> Bob Hanlon
>
>
> On Fri, Aug 31, 2012 at 3:59 AM,  <"Andreas Talmon l'Arm=E9e"@smc.vnet.net> wrote:
>> Hi All
>>
>> Is there a way to tell mathematica to solve only for real solutions. My
>> differential equation is of the kind
>>
>> y''''[x]+a y[x]==0
>>
>> a= constant coefficient
>>
>> I know that I get 4 komplex eigenvalues which are complex conjungated.
>> But y[x] is a real function.
>> Solving this equation with DSolve always gets a complex function y[x].
>>
>> Any Ideas.
>>
>> Thanks, Andreas
>>

• References:
  • Re: DSolve for a real function
    • From: Bob Hanlon <hanlonr357@gmail.com>