Re: newbie question DSolve

• To: mathgroup at smc.vnet.net
• Subject: [mg52090] Re: newbie question DSolve
• From: "Dr. Wolfgang Hintze" <weh at snafu.de>
• Date: Wed, 10 Nov 2004 04:45:41 -0500 (EST)
• References: <cmppui\$mll\$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```If you replace DiracDelta[d - x]*y[x] by the equivalent DiracDelta[d -
x]*y[d] then your equation can be solved as follows (with just one minor
error message appearing twice, which can be ignored)

In[1]:=
s = DSolve[{-y[x] + Derivative[2][y][x] == DiracDelta[d - x]*y[d], y[0]
== 0, y[L] == 0}, y[x], x]

From In[1]:=
DSolve::"nvld" : "The description of the equations appears to be
ambiguous or \
invalid."

Out[1]=
{{y[x] -> 1/2*E^(-d - x)*(-((E^(2*x)*((-1 + E^(2*d))*UnitStep[-d] -
(E^(2*d) - E^(2*L))*UnitStep[-d + L])*y[d])/
(-1 + E^(2*L))) + ((E^(2*L)*(-1 + E^(2*d))*UnitStep[-d] -
(E^(2*d) - E^(2*L))*UnitStep[-d + L])*y[d])/
(-1 + E^(2*L)) - E^(2*d)*UnitStep[-d + x]*y[d] +
E^(2*x)*UnitStep[-d + x]*y[d])}}

From In[2]:=
DSolve::"nvld" : "The description of the equations appears to be
ambiguous or \
invalid."

Extracting the solution to u[x]

In[4]:=
u[x_] = y[x] /. s[[1]]

you can Plot it, after assigning numeric values to all relevant quantities:

In[6]:=
L = 1; d = 0.5; y[d] = 1;
Plot[u[x], {x, -1, 4}, PlotRange -> {{-1, 5}, {-2, 1}}];

Hope this hepls
Wolfgang

Pratik Desai wrote:

> Hello all
>
> I am trying to use DSolve to solve a ode with discontinuity in it (wave
> equation with a viscous damper injected at a location d)
>
> This is what i am using
>
> DSolve[{y''[x]-lamda^2*y[x]==DiracDelta[x-d]*y[x],y[0]==0,y[L]==
> =0},y[x],x]
>
> the problem I am facing is that
>
> y[x] on the right hand side (next the delta function) varies w.r.t to
> the location
>
> y[x]==y[x]&& 0<=x<=d
> y[x]==y[L-x]&&d<=x<=L
>
> I can solve the above equation without the y[x] coupled to the delta
> function
>
>
>
> Pratik Desai
>
>
> ps: This is my third attempt at posting my query, I hope this time it
> makes it to the list :)
>
>
>
>

```

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