Types of equations that can be solved algebraically.

D

Dale10101

Full Member
Joined
Feb 25, 2013
Messages
318
[FONT=Verdana, Arial, Helvetica, sans-serif]I understand that some equations like sin(x) = log(x) +x^2 cannot be solved algebraically ... must be solved numerically. Can one know which equations can NOT be solved on their face. Is it a matter of incommensurate domains? Or, is it simply that no one has yet devised a means of transforming one type of function into another? In any case is there perhaps a table that lists which functions can be combined in an equation with at least the possibility of finding a solution.

I know that the general question of which equations have a solution or even what is meant by a solution is a very deep subject. Mostly I am interested what type of the more mundane functions can and cannot be combined and explicitly solved such as the one I listed, and perhaps why in a hand waving sort of way. Thanks.[/FONT]
 
Dale10101 said:
I understand that some equations like sin(x) = log(x) +x^2 cannot be solved algebraically ... must be solved numerically. Can one know which equations can NOT be solved on their face. Is it a matter of incommensurate domains? Or, is it simply that no one has yet devised a means of transforming one type of function into another? In any case is there perhaps a table that lists which functions can be combined in an equation with at least the possibility of finding a solution.

I know that the general question of which equations have a solution or even what is meant by a solution is a very deep subject. Mostly I am interested what type of the more mundane functions can and cannot be combined and explicitly solved such as the one I listed, and perhaps why in a hand waving sort of way. Thanks.
Click to expand...

In general, nothing beyond 4 th power polynomial can be solved in "closed form".

sin(x), log(x), cos(x) are actually representation of infinite series. Any general function involving those cannot be solved in closed form. Some particular representations can be solved (in closed form).

for example:

We can solve in closed form the following equation:

sin(x) - √2/2 = 0 → x = π/4

however we cannot solve in closed form, the following,

sin(x) - √2/π = 0 → x = ??
 
Last edited by a moderator:
Thank you

I am thinking about what you wrote and considering how the transcendental functions are interrelated i.e. the already established transformation forumuli, also, researching the conceptual underpinnings of how one finds the correct series representation for a particular function, first, however, more review of basic textbook problems, was just trying to get a peek ahead. Thanks.
 
Dale10101 said:
I am thinking about what you wrote and considering how the transcendental functions are interrelated i.e. the already established transformation forumuli, also, researching the conceptual underpinnings of how one finds the correct series representation for a particular function, first, however, more review of basic textbook problems, was just trying to get a peek ahead. Thanks.
Click to expand...

Taylor's Series
 
You must log in or register to reply here.
Top