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Re: simple question on DSolve

• To: mathgroup at smc.vnet.net
• Subject: [mg124972] Re: simple question on DSolve
• From: Barrie Stokes <Barrie.Stokes at newcastle.edu.au>
• Date: Wed, 15 Feb 2012 04:42:06 -0500 (EST)
• Delivered-to: l-mathgroup@mail-archive0.wolfram.com
• References: <201202141137.GAA17715@smc.vnet.net>

Hi Gualtiero

It's always helpful to see your code - there's many a slip between cup and lip.

If someone had shown me this before I saw your email, I wouldn't have suspected anything was wrong. BTW, I am *no* DE guru at all (hence my somewhat pedestrian code below); my last DE text book was "Differential Equations" by Ralph Palmer Agnew, 2nd Ed., 1960.

(I wonder what he would have thought of Mathematica. There's a very nice obit at http://ecommons.cornell.edu/bitstream/1813/19237/1/Agnew_Ralph_Palmer_1986.pdf)

In[1]:= ClearAll[ a, b, x, y, z ]

In[2]:= sol00 = 
 DSolve[ {y''[x] + k y[x] == 0, y'[0] == 0, y'[1] == 0}, y[x], x] // 
  First

Out[2]= {y[x] -> 0}

In[3]:= sola0  = 
 DSolve[ {y''[x] + k y[x] == 0, y'[0] == a, y'[1] == 0}, y[x], x] // 
   First // Simplify

Out[3]= {y[x] -> (a Cos[Sqrt[k] (-1 + x)] Csc[Sqrt[k]])/Sqrt[k]}

In[4]:= Limit[ y[x] /. sola0, a -> 0 ] // Simplify

Out[4]= 0

In[5]:= sol0b = 
 DSolve[ {y''[x] + k y[x] == 0, y'[0] == 0, y'[1] == b}, y[x], x] // 
   First // Simplify

Out[5]= {y[x] -> -((b Cos[Sqrt[k] x] Csc[Sqrt[k]])/Sqrt[k])}

In[6]:= Limit[ y[x] /. sol0b, b -> 0 ] // Simplify

Out[6]= 0

In[7]:= solab = 
 DSolve[ {y''[x] + k y[x] == 0, y'[0] == a, y'[1] == b}, y[x], x] // 
   First // Simplify

Out[7]= {y[
   x] -> ((a Cos[Sqrt[k] (-1 + x)] - b Cos[Sqrt[k] x]) Csc[Sqrt[k]])/
  Sqrt[k]}

In[8]:= Limit[ y[x] /. solab, b -> 0 ] // Simplify

Out[8]= (a Cos[Sqrt[k] (-1 + x)] Csc[Sqrt[k]])/Sqrt[k]

In[9]:= Limit[ y[x] /. solab, a -> 0 ] // Simplify

Out[9]= -((b Cos[Sqrt[k] x] Csc[Sqrt[k]])/Sqrt[k])

I hope this helps.

Cheers

Barrie

>>> On 14/02/2012 at 10:37 pm, in message <201202141137.GAA17715 at smc.vnet.net>,
Gualtiero Badin <gualtiero.badin at gmail.com> wrote:
> Hello,
> if I try to solve the simple boundary value problem
> 
> y''+ky=0
> y'(0)=0
> y'(1)=0
> 
> mathematica returns me y=0, that is correct but that is not the
> complete answer... Does anyone know how to get the complete answer?
> (of course i know the complete answer, but I would like to solve some
> uglier versions of the same problem...)
> Thanks

• References:
  • simple question on DSolve
    • From: Gualtiero Badin <gualtiero.badin@gmail.com>