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Re: DSolve Bessels


Alberto Verga wrote:

> Mathematica 3 seems to be not able to show that J_1(x) is solution of
> the Bessel equation:
>
> in: yy=BesselJ[1,x]
>
> in: Simplify[D[yy,{x,2}]+D[yy,x]/x+(1-1/x^2)yy]
>
> out: 1/(4x^2) (x^2 BesselJ[-1, x] + 2 x BesselJ[0, x] - 4 BesselJ[1, x]
> +
>       2 x^2 BesselJ[1, x] - 2 x BesselJ[2, x] + x^2 BesselJ[3, x])
>
> Using trivial transformations one gets 0, Mathematica does it not.
> One obtains the correct answer (out: 0) in other systems.

You need to use FullSimplify (or Simplify and FunctionExpand) in 3.0 or 4:

Mathematica 3.0 for Digital Unix

In[1]:= FullSimplify[(1 - 1/x^2)*BesselJ[1, x] +
    D[BesselJ[1, x], x]/x + D[BesselJ[1, x], {x, 2}]]

Out[1]= 0

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