Solving Math Problems: Trial & Error vs. Other Methods

[FeDeX_LaTeX]

Hello;

Is there any type of mathematical problem which can only be solved by trial and error (and therefore no other method has been found)? For example, for a cubic equation x^3 + x = 25, one could use trial and error, but a method arrives at the answer too.

Thanks.

 

[HallsofIvy]

Well, exactly what do you mean by a "method"? The equation \(\displaystyle xe^x= 1\) can be solved by the "Lambert W function", x= W(1), precisely because the Lambert W function is defined as the inverse function to \(\displaystyle f(x)= xe^x\). But how would you evaluate that? For that matter if found that the solution to a different equation were \(\displaystyle x= e^{\pi}\), how would you evaluate that? (Using a calculator, of course- and how does the calculator find the value?)

The fact is that almost all equations must be solved by numerical methods and numerical methods are often variations of "educated trial and error". There have been a few threads on this board on the "mid-point" or "bisection" method of solving equations or "Newton's method", which are in essense "trial and error"- you evaluate f(x) and see if it is equal to the value you want. The "educated" part is that you can use how f(x) differs from the desired value to make your next "trial".

 

[yuiop]

Try solving x + sin(x) = y for x.