Maxima can't solve this integral equation

Okay, I got it now.

Thanks,

Rich


 ------------Original Message------------
From: "Richard Hennessy"<rvh2007 at comcast.net>
To: "Raymond Toy" <toy.raymond at gmail.com>
Cc: "Maxima List" <maxima at math.utexas.edu>
Date: Wed, Jun-25-2008 11:48 PM
Subject: Re: [Maxima] Maxima can't solve this integral equation

Sorry,

Lets start over.

(%i1) assume(E[n]>0,mu>0,m>0);

The first equation is an integral equation.

(%i2) integrate(sqrt(2*m*(E[n]-mu*x^4)),x,-x[0],x[0])=(n-1/2)*%pi*hbar;

substituting

(%i3) subst(x[0]=(E[n]/mu)^(1/4),%);

you get then 

(%o4) integrate(sqrt(2*m*(E[n]-mu*x^4/E[n])),x,-(E[n]/mu)^(1/4),(E[n]/mu)^(1/4))=(n-1/2)*%pi*hbar;

using changevar you get

(%i5) changevar(%o4,z-(mu/E[n])^(1/4)*x, z, x);

(%o5) (sqrt(2)*sqrt(m)*E[n]^(1/4)*integrate(sqrt(E[n]-z^4),z,-1,1))/mu^(1/4)=%pi*hbar*(n-1/2)

solving for E[n],

(%i6) solve([%], [E[n]])[1];

gives 

(%o6) integrate(sqrt(E[n]-z^4),z,-1,1)=(%pi*hbar*mu^(1/4)*(2*n-1))/(2*sqrt(2)*sqrt(m)*E[n]^(1/4))

which is not solved for E[n] because I can't get that E[n] out of the integral.  According to the text I should get.

E[n] = (%pi/C*mu^(1/4)*hbar/sqrt(2*m)*(n-1/2))^(4/3)

where C = integrate(sqrt(1-z^4),z,-1,1);

I can't make that last transform in Maxima.

Rich

ps I'll take your really hackish code for teaching Maxima if you can provide it and it can do this one.





 ------------Original Message------------
From: Raymond Toy <toy.raymond at gmail.com>
To: "Richard Hennessy" <rvh2007 at comcast.net>
Cc: "Maxima List" <maxima at math.utexas.edu>
Date: Wed, Jun-25-2008 4:44 PM
Subject: Re: [Maxima] Maxima can't solve this integral equation

Richard Hennessy wrote:
> Hi,
>
> (%i3) integrate(sqrt(2*m*(E[n]-mu*x^4)),x,-a[0],a[0])=(n-1/2)*%pi*hbar
>
> It gives just the noun form.
>
> But if you do the following substitution manually it can be done.
>
> z=(mu/E[n])^(1/4)*x
>
> You should get an answer after solving for E[n] of
>
> E[n] = %pi/C*mu^(1/4)*hbar/sqrt(2*m)*(n-1/2)
>
> where C can be computed numerically from 
>
> (%i4) quad_qags(sqrt(1-z^4), z, -1, 1);
>
> Is this a bug or is it the way I am formulating the problem?
>
>   
Not sure if it's a bug, but there is some information missing.  I don't 
see how, from your original integral with limits +/- a[0], you can use 
your substitution to get an integral from -1 to 1.  In any case your 
integral is an elliptic integral, and  Maxima doesn't know anything 
about elliptic integrals.  (I have some really hackish code to teach 
maxima a little about elliptic integrals, but it's not very complete nor 
well thought out.)

Ray




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