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From: Jason Quinn on 12 Jun 2010 05:30
Suppose I have an expression of the form

1/2 = int_0^1 besselI[0,x*t] dt

and I want to find the value of "x" that will make the integral true.
Can Mathematica handle such situations? I've tried all the main
suspects that I get warnings that the expression depends on x in a non-
algebraic way. The numerical tasks do not seem to work either. The
expression above is just a made-up example, but in general, what do to
when you are trying to solve equations of this type? I could
certainly write a function that find the value iteratively myself, but
I'd be surprised if such a thing doesn't already exist.

Cheers,
Jason

PS I've tried Wolfram Alpha and

solve 1 = int besselI0(x*t) from t=0 to 1

fails but

solve 1/2 = int erf(x*t) from t=0 to 1

works.

From: Bill Rowe on 13 Jun 2010 04:13
On 6/12/10 at 5:30 AM, jason.lee.quinn(a)gmail.com (Jason Quinn) wrote:

>Suppose I have an expression of the form

>1/2 = int_0^1 besselI[0,x*t] dt

>and I want to find the value of "x" that will make the integral
>true. Can Mathematica handle such situations? I've tried all the
>main suspects that I get warnings that the expression depends on x
>in a non- algebraic way. The numerical tasks do not seem to work
>either. The expression above is just a made-up example, but in
>general, what do to when you are trying to solve equations of this
>type? I could certainly write a function that find the value
>iteratively myself, but I'd be surprised if such a thing doesn't
>already exist.

For a numeric solution to non-algebraic expressions try
FindRoot. However, for your specific example, there doesn't
appear to be any solution. That is:

In[8]:= int = Integrate[BesselI[0, x t], {t, 0, 1}]

Out[8]= HypergeometricPFQ[{1/2}, {1, 3/2}, x^2/4]

In[9]:= FindMinimum[int, x]

Out[9]= {1.,{x->-7.63245*10^-12}}

And note for solving equations with an integral in them, it is
almost always better to first solve the integral then solve for
the solution. This ensures the integral is not evaluated
repeatedly needlessly.


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