[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>