- #1
Cheman
- 235
- 1
Does anyone know how to solve the following equation:
e^x-x^3+x^2-2x = 0.
Thanks in advance.
e^x-x^3+x^2-2x = 0.
Thanks in advance.
How Do You Solve the Equation e^x - x^3 + x^2 - 2x = 0?
- Context: Undergrad
- Thread starter Cheman
- Start date
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Discussion Overview
The discussion revolves around solving the equation e^x - x^3 + x^2 - 2x = 0. Participants explore various methods for finding solutions, including numerical approximations and graphical methods, while also addressing the nature of the equation as transcendental.
Discussion Character
- Exploratory
- Technical explanation
- Mathematical reasoning
- Debate/contested
Main Points Raised
- Some participants provide approximate solutions, noting values around x = 1.727780... and x = 4.098178...
- One participant emphasizes that the equation is transcendental and suggests that solutions can only be found through approximation, recommending graphical methods.
- A participant reiterates the equation and discusses formatting options for displaying mathematical expressions in the forum.
- Another participant describes a method of isolating x and making iterative guesses to approach a solution, detailing a sequence of approximations that converge to the first solution.
- It is noted that the method described may not yield the second solution unless certain conditions about the slope of the function are met, and an alternative formulation is suggested for finding the second solution, albeit with slower convergence.
Areas of Agreement / Disagreement
Participants generally agree that the equation is transcendental and that approximation methods are necessary for finding solutions. However, there are multiple approaches discussed, and the effectiveness of these methods remains contested.
Contextual Notes
Participants mention limitations regarding the convergence of certain methods and the dependence on the slope of the function in the region of the solution. The discussion does not resolve these limitations.
- #2
arcnets
- 493
- 0
x = 1.727780...
x = 4.098178...
x = 4.098178...
- #3
dextercioby
Science Advisor
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It doesn't work that way...It's a transcendental equation.It's solutions are found only by approximation...Graphical method is the best to use...
Daniel.
Daniel.
- #4
NeutronStar
- 419
- 1
Cheman said:
Do you know that without using LaTex you can still do superscripts using the [ sup ] [ /sup ] tags (without spaces)
For example. e^x can be written as e[ sup ]x[ /sup ] (to produce) ex.
Type in [ sup ] (without the spaces), copy it, then just paste it as you type. Then add the slash to the trailing tags.
With just a little pasting your equation could look like this,...
ex-x3+x2-2x = 0.
Or even encapsulate the whole mess in a larger font size llike so,...
ex-x3+x2-2x = 0
The're even nestable. For example you could do something like the following without using latex at all.
e(x2-y2)
You can also do subscripts the same way using [ sub ] [ /sub ]
Sorry. I just hate messy looking equations.
- #5
dextercioby
Science Advisor
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The good part is that,even written in this "horrible" way,it's still inteligible.Thankfully it didn't involve fractions... 
Daniel.
Daniel.
- #6
NeutronStar
- 419
- 1
Yep, that's what I got too. These calculators never cease to amaze me.arcnets said:
- #7
Integral
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One way to come up with a solution for this type of problem is to simply to isolate an x on one side of the equation. Then make a guess, following guesses are made by simply recomputing using the most recent result.
for example in this problem you can write
[tex]x = \frac { e^x -x^3+x^2} 2[/tex]
With a starting guess of 1 the following sequence is generated
2.718281828
1.228890709
1.535886038
1.69065678
1.724461477
1.72755973
1.727765576
1.727778787
1.727779633
1.727779687
1.727779691
Trouble is this particular formulation will not generate the second point. It is required that the slope of the function be less then 1 in the region of the solution.
if you recast as
[tex]x= ln (x^3 -x^2 +2)[/tex]
You get a fixed point at the second solution but is much slower in converging.
This process is know as a Fixed point solution. A web search may well provide more details.
for example in this problem you can write
[tex]x = \frac { e^x -x^3+x^2} 2[/tex]
With a starting guess of 1 the following sequence is generated
2.718281828
1.228890709
1.535886038
1.69065678
1.724461477
1.72755973
1.727765576
1.727778787
1.727779633
1.727779687
1.727779691
Trouble is this particular formulation will not generate the second point. It is required that the slope of the function be less then 1 in the region of the solution.
if you recast as
[tex]x= ln (x^3 -x^2 +2)[/tex]
You get a fixed point at the second solution but is much slower in converging.
This process is know as a Fixed point solution. A web search may well provide more details.
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