How to solve "advanced" equation?

[Petanquell]

Hey guys,

 

I've never got problem with solving quadratic, logarithmic or goniometric equations, but I don't know how to solve them when they are all together.

The most simple example is:

[math] \textup{log}(x)+\textup{sin}(x)+x^2+x+1=0 [/math]

Is there any way to solve them without "computer help"?

 

Thanks, pq

[Bignose]

You could graph it to get a good idea of where the roots are. But, no, in general when you start getting nonlinear, then analytical solutions are going to be rarer and rarer. That's the primary motivation behind the powerful computational techniques that we have today.

[D H]

The most simple example is:

[math] \textup{log}(x)+\textup{sin}(x)+x^2+x+1=0 [/math]

Is there any way to solve them without "computer help"?

Sure. You could do exactly what a computer does, by hand. The word computer is old. Computers were people. Gauss employed a whole army of computers to do numerical computations for him.

 

The function at hand obviously has a zero somewhere between x=0 and x=1 as log(x) tends becomes unbounded negative as x->0 and all of the other terms are positive over (0,1). It has exactly one zero between 0 and 1 because the four varying terms are all monotonically increasing over 0 and 1. Finally, this is the only zero. For x>1, log(x), x², x, and 1 are all positive, so the only places the function can be zero is where sin(x) is negative. As sin(x) >= -1, 1+sin(x) >= 0. As log(x), x, and x² are all positive for x>1, there are no zeros for this function beyond x=1.

 

Bottom line: The function has exactly one zero located somewhere between 0 and 1, with no number crunching whatsoever.

[ydoaPs]

You could try Wolfram Alpha.