How can I solve the equation x-esin(x)=4?

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In summary, The conversation is about solving an equation analytically or symbolically, specifically one involving Euler's number and a constant. The expert explains that there is no closed form solution for this type of equation and mentions the use of functions like the Lambert W function. They also mention that in some cases, there may be analytic solutions, but not in general. The conversation ends with the person understanding why others have tried different methods to solve these equations.
  • #1
PainterGuy
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Hi,

How can I solve the following equation analytically or symbolically? "e" is Euler's number. Instead of "4", you could use the letter "c" for a constant. Could you please help me? Thank you.

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  • #2
You can't.

The long answer: There is no closed form for solutions to equations like that. The same problem appears for x+ex=c and similar. There are functions defined as "let y be the solution to this problem" (like the Lambert W function, you can express x in terms of these functions, but that is not really an answer.

In some cases there are analytic solutions that can be found (e.g. x+3sin(x)=0 has a trivial solution), but that doesn't work in general.
 
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  • #3

1. How do I find the value of x in the equation x-esin(x)=4?

To solve this equation, you will need to use a numerical or graphical method. One common method is to use a graphing calculator to plot both sides of the equation and find the intersection point, which will give you the value of x. Another method is to use an iterative approach, where you make an initial guess for x and then use a series of calculations to refine your guess until you reach a solution.

2. Can I use algebra to solve the equation x-esin(x)=4?

No, this equation cannot be solved algebraically because it contains both a variable (x) and a trigonometric function (sin(x)). Algebraic methods can only be used for equations that contain variables and basic arithmetic operations.

3. What is the domain of the equation x-esin(x)=4?

The domain of this equation is all real numbers. Since there are no restrictions on the variable x, any real number can be substituted into the equation.

4. Are there any special techniques or formulas for solving this type of equation?

Yes, there are several techniques and formulas that can be used to solve equations involving trigonometric functions. Some common methods include using trigonometric identities, substitution, or approximations. It is important to choose the method that best suits the specific equation and the available tools.

5. How can I check if my solution to the equation x-esin(x)=4 is correct?

You can check your solution by plugging it back into the original equation and seeing if it satisfies the equation. If the solution is correct, both sides of the equation will be equal. You can also use a graphing calculator to plot the equation and your solution, and see if they intersect at the point (x,4).

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