Roots of implicit equations

tom92373

I was working on double integrals when I came across the equation: x^(3/2)=sin(x).
There was no noticeable way to isolate the equation for x without having a function of x equal to x. I am wondering how to isolate equations involving logs, sines, etc when it is given in an implicit form.
Using a computer, I was able to get an approximation of 0 and 8.02... How do I get the EXACT value of x?

qntty

I don't think that can be solved analytically, numerical approximations are the best you can get.

CRGreathouse

There's no reason to expect a nice solution to that equation.

To 500 decimal places:
0.802803731737889315511835324604000441222668910616527410810139645656916 41862577997739822547061430396268572323604994666281323668533410644604205 80146429193050351847866748648721823651393578239737490947961432790796313 11192258789712012684896470290853854071877856944549231720563315930180837 75727247023723969536341968998158469732909155080566871504200160137298683 45016085397258496851256650987721510001930807383556524999088268285074848 68972435998828725360089377601379653239348761648787005801149203560836827 42718

DuncanM

How did you get an answer to 500 decimal places?!

adriank

Mathematica (for example) can do it.

arildno

What's wrong with x=0 as an exact solution??