How to Isolate Implicit Equations Involving Logs, Sines, and More?

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The discussion centers on the challenge of isolating implicit equations involving functions like logs and sines, specifically the equation x^(3/2) = sin(x). The user seeks to find the exact value of x, noting that numerical approximations yield values around 0 and 8.02, but they doubt an analytical solution exists. They mention achieving a highly precise approximation of x to 500 decimal places using computational tools like Mathematica. The conversation also touches on the validity of x=0 as an exact solution, prompting further inquiry into the nature of implicit equations. Overall, the consensus leans towards the difficulty of finding exact solutions for such equations, with numerical methods being the most viable approach.
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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?
 
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I don't think that can be solved analytically, numerical approximations are the best you can get.
 
There's no reason to expect a nice solution to that equation.

To 500 decimal places:
0.80280373173788931551183532460400044122266891061652741081013964565691641862577997739822547061430396268572323604994666281323668533410644604205801464291930503518478667486487218236513935782397374909479614327907963131119225878971201268489647029085385407187785694454923172056331593018083775727247023723969536341968998158469732909155080566871504200160137298683450160853972584968512566509877215100019308073835565249990882682850748486897243599882872536008937760137965323934876164878700580114920356083682742718
 
How did you get an answer to 500 decimal places?!
 
Mathematica (for example) can do it.
 
What's wrong with x=0 as an exact solution??
 

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