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[DSolve] Re: newbie question DSolve (revisited)

• To: mathgroup at smc.vnet.net
• Subject: [mg52224] [DSolve] Re: newbie question DSolve (revisited)
• From: Paul Abbott <paul at physics.uwa.edu.au>
• Date: Mon, 15 Nov 2004 03:17:32 -0500 (EST)
• Organization: The University of Western Australia
• References: <cmppui$mll$1@smc.vnet.net> <200411100945.EAA11259@smc.vnet.net> <cmve9e$shc$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

In article <cmve9e$shc$1 at smc.vnet.net>,
 "Pratik Desai" <pdesai1 at umbc.edu> wrote:

> I tried your suggestion unfortunately, Mathematica gives me another error it 
> is as follows
> 
> In[10]:=
> DSolve[{y''[x] - lamda^2*y[x] == DiracDelta[x - d]*y[d], y[0] == 0,
> y[l] == 0}, y[x], x]
> 
>   (DSolve::"litarg"),  "To avoid possible
>     ambiguity, the arguments of the    dependent variable in (the equation) 
> should literally match the independent variables".

As others have suggested, try solving 

  y''[x] - lamda^2 y[x] == DiracDelta[x - d] yd

and then determining a self-consistent solution at x = d. My analysis of 
this problem find only the trivial solution y[x] = 0 for 0 <= x <= l for 
real lambda. However, for imaginary lambda, there are nontrivial  
solutions (i.e., eigenvalues).

[1] Solve the differential equation:

  sol = First[DSolve[{y''[x] - lambda^2 y[x] == DiracDelta[x - d] yd, 
    y[0] == 0, y[l] == 0}, y, x]];

[2] Determine the self-consistent solutions.

 consistent = Simplify[y[d] == (y[d] /. sol /. yd -> y[d]), 0 < d <= l]];

[3] Not suprisingly, Solve only yields the trivial solution.

  Solve[consistent, y[d]]

[4] To be an eigenvalue the following expression must vanish:

  eval = Simplify[Factor[Subtract @@ consistent]/y[d]]

[5] A plot of eval for, say, l -> 1 and d -> 1/2, shows that it never 
vanishes. However, putting lambda -> I lambda and simplifying,

  complexeval = Simplify /@ ComplexExpand[eval /. lambda -> I lambda]

one _can_ find eigenvalues by numerically solving the resulting 
transcendental equation. For example, again with l -> 1 and d -> 1/2, 

  Simplify[complexeval /. {l -> 1, d -> 1/2}]

  FindRoot[%, {lambda, 4}]

Cheers,
Paul

-- 
Paul Abbott                                   Phone: +61 8 6488 2734
School of Physics, M013                         Fax: +61 8 6488 1014
The University of Western Australia      (CRICOS Provider No 00126G)         
35 Stirling Highway
Crawley WA 6009                      mailto:paul at physics.uwa.edu.au 
AUSTRALIA                            http://physics.uwa.edu.au/~paul

• References:
  • Re: newbie question DSolve
    • From: "Dr. Wolfgang Hintze" <weh@snafu.de>