Solving a transcendental equation

In summary, the conversation discusses the equation x^2*(log x)^{1/3}=-C and its asymptotic solution for very small C. It is suggested that the book has a typo in its formula, missing the power 1/3 in the coefficient 2^(1/3). A better approximation for the solution is also presented in an attachment.
  • #1
geoduck
258
2
Suppose you have the equation

[tex]x^2*(log x)^{1/3}=-C [/tex]

for very small C. I have a book claims that asymptotically, for very small C, the solution is:

[tex]x^2=C*\frac{2}{[logC]^{1/3} } [/tex]

I'm not quite sure how to show this. If the 2 wasn't there, it looks like what they did was:

[tex]x^2=-\frac{C}{(log x)^{1/3}} [/tex]

and just substituted -C for x on the RHS:

[tex]x^2=\frac{C}{(log C)^{1/3}} [/tex]

I was thinking maybe you could substitute -C for x2 instead to get:[tex]x^2=\frac{2^{1/3}C}{(log C)^{1/3}} [/tex]
 
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  • #2
I agree, there is a typo in the formula of the book : the power 1/3 is missing in the coefficient 2^(1/3)
An even better approximate can be derived (in attachment)
 

Attachments

  • Approximate.JPG
    Approximate.JPG
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Related to Solving a transcendental equation

1. What is a transcendental equation?

A transcendental equation is an equation that involves transcendental functions, such as logarithms, exponentials, trigonometric functions, or their inverses. These equations cannot be solved algebraically and require numerical methods to find their solutions.

2. How do you solve a transcendental equation?

There are several numerical methods for solving a transcendental equation, such as the bisection method, Newton's method, or the secant method. These methods involve making an initial guess for the solution and then using iterative calculations to get closer and closer to the actual solution.

3. Can all transcendental equations be solved?

No, not all transcendental equations have closed-form solutions that can be expressed algebraically. In some cases, numerical methods are the only way to find the solutions. Additionally, some transcendental equations may have infinite solutions, making them impossible to solve.

4. What are some real-life applications of solving transcendental equations?

Transcendental equations are commonly used in physics, engineering, and other scientific fields to model real-world phenomena. For example, the logarithmic function is often used to model exponential growth, while the sine and cosine functions are used to model periodic motion.

5. Are there any tips for solving transcendental equations?

One tip is to try different numerical methods if one method does not converge to a solution. It is also helpful to have an understanding of the behavior of transcendental functions, such as their graphs and properties, to make educated guesses for initial solutions. Additionally, using a graphing calculator or software can be helpful in visualizing the solutions and checking for accuracy.

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