How to Isolate Implicit Equations Involving Logs, Sines, and More?

In summary, the conversation discusses the difficulty of isolating equations involving logs, sines, and other functions when they are given in implicit form. The speaker was able to get an approximation of 0 and 8.02 using a computer, but is wondering how to obtain the exact value of x. However, it is believed that there may not be an analytical solution and numerical approximations are the best option. The conversation also mentions that tools like Mathematica can provide answers to a high degree of accuracy, and questions why the exact solution of x=0 is not considered sufficient.
  • #1
tom92373
3
0
I was working on double integrals when I came across the equation: x^(3/2)=sin(x).
There was no noticeable way to isolate the equation for x without having a function of x equal to x. I am wondering how to isolate equations involving logs, sines, etc when it is given in an implicit form.
Using a computer, I was able to get an approximation of 0 and 8.02... How do I get the EXACT value of x?
 
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  • #2
I don't think that can be solved analytically, numerical approximations are the best you can get.
 
  • #3
There's no reason to expect a nice solution to that equation.

To 500 decimal places:
0.80280373173788931551183532460400044122266891061652741081013964565691641862577997739822547061430396268572323604994666281323668533410644604205801464291930503518478667486487218236513935782397374909479614327907963131119225878971201268489647029085385407187785694454923172056331593018083775727247023723969536341968998158469732909155080566871504200160137298683450160853972584968512566509877215100019308073835565249990882682850748486897243599882872536008937760137965323934876164878700580114920356083682742718
 
  • #4
How did you get an answer to 500 decimal places?!
 
  • #5
Mathematica (for example) can do it.
 
  • #6
What's wrong with x=0 as an exact solution??
 

1. What are implicit equations?

Implicit equations are mathematical equations that relate one or more variables together. Unlike explicit equations, where one variable is expressed in terms of other variables, implicit equations do not explicitly state the relationship between variables. This means that the equation cannot be solved for any one variable, but rather describes a relationship between the variables.

2. What are the roots of an implicit equation?

The roots of an implicit equation are the values of the variables that make the equation true. These values satisfy the relationship described by the equation and can be found by solving the equation for one variable in terms of the others. In some cases, there may be multiple sets of roots that satisfy the equation.

3. How are implicit equations used in science?

Implicit equations are used in various fields of science, including physics, biology, and engineering. They are often used to describe complex relationships between variables that cannot be easily expressed in explicit equations. These equations are also used in mathematical modeling to represent real-world phenomena and make predictions about their behavior.

4. How can we graph implicit equations?

Graphing implicit equations can be done by plotting points on a coordinate plane that satisfy the equation. Since these equations do not explicitly define one variable in terms of the others, it may be necessary to use a computer or calculator to generate a large number of points to accurately represent the relationship. Alternatively, implicit equations can also be graphed by converting them to explicit form.

5. What is the difference between implicit and explicit equations?

The main difference between implicit and explicit equations is that explicit equations explicitly define the relationship between variables, while implicit equations do not. This means that explicit equations can be solved for a specific variable, while implicit equations cannot. Implicit equations are often used to describe more complex relationships, while explicit equations are used for simpler relationships.

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