Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [transcendental-equations]

Transcendental equations are equations containing transcendental functions, i.e. functions which are not algebraic. An algebraic function is a function that satisfies a polynomial equation whose terms are themselves polynomials with rational coefficients.

Transcendental equations are equations containing transcendental functions, i.e. functions which are not algebraic. An algebraic function is a function that satisfies a polynomial equation whose terms are themselves polynomials with rational coefficients.

448 questions
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How to solve $\upsilon^\upsilon=\upsilon+1$

What is the real positive $\upsilon$ that satisfies $\upsilon^\upsilon=\upsilon+1$? I think the Lambert-W function might be relevant here, but I have no idea how to use it. $\upsilon\approx 1.775678$ I just really like the letter upsilon. It doesn't…
B H
  • 1,424
10
votes
4 answers

How do I solve this exponential equation? $5^{x}-4^{x}=3^{x}-2^{x}$

How do I solve this exponential equation? $$5^{x}-4^{x}=3^{x}-2^{x}$$
Young
  • 5,492
7
votes
4 answers

The roots of equation $x3^x=1$ are

I have to find roots of equation $x3^x=1$ A.Infinitely many roots B.$2$ roots C.$1$ root D. No roots\ How do i start? Thanks
Sophie Clad
  • 2,291
7
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4 answers

Solving the equation $x e^x = e$

I know that $x e^x = e$ means $x = 1$, but how do you solve for it?
Perfecto Bautista
4
votes
2 answers

$\log_j(\log_j(\log_j(x)))=\log(x);\ \ j=?$

$\log_j(\log_j(x))=\log(x)$ has solution $j=x^{\exp-W(\log^2(x))}$ for real $x\neq0$, where $W=$ Lambert W function. But what is the solution to $\log_j(\log_j(\log_j(x)))=\log(x)$? Mathematica can't do it - can it be done?
martin
  • 8,998
4
votes
3 answers

How to find number of real roots of a transcendental equation?

The number of real roots of the equation $$2\cos\left(\frac{x^2+x}6\right)=2^x+2^{-x}$$ Another question is... can we use descartes rule of sign in here or in any transcendental equation ?
Sourav
  • 281
4
votes
1 answer

Transcendental equation

If $\alpha$ be a root of $x^{x-\sqrt{x}} = \sqrt{x}+1$. We need to find the value(s) of $(\alpha + \frac{1}{\alpha})$. WolframAlpha shows two possible roots of the equation.
Aroonalok
  • 394
3
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3 answers

May $y=e^x$ be satisfied with both $x$ and $y$ $\in$ $\mathbb{Z}^+$?

May $y=e^x$ be satisfied with both $x$ and $y$ be positive integers? I think it is not possible as $e$ ,a transcendental number, when multiplied by itself would never result in rational number. Am I right?
kaka
  • 1,896
3
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3 answers

Roots of transcendent equations $\tan{x}=bx$ and $x\tan{x}=b$

We know that transcendent equations $$\tan{x}=bx$$ and $$x\tan{x}=b$$ can not be solved exactly. But what I concerned most is the relationship between their non-trival roots $x_{n}^{(1)}$ and $x_{n}^{(2)}$, where $x_{n}^{(1)}$ and $x_{n}^{(2)}$…
Roger209
  • 1,233
3
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Special functions useful for solving a transcendental equation

I have the following transcendental equation which may very well lack an analytic solution. I would, at the least, like an expression for the relationship between $\theta$ and $\phi$ in some closed…
Michael Jarret
  • 331
3
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3 answers

A formal way to solve a transcendental equation

Is there a formal way of solving the equation $$x^x = \frac{1}{\sqrt{2}}\ ?$$The solutions are $x = \frac{1}{4},\frac{1}{2}$. This can be easily obtained by plotting the function or just by guessing the solutions. However, is there a general…
user54031
2
votes
1 answer

How does one analyze $\phi = \beta \sin\phi$?

Consider the following transcendental equation: $$\phi = \beta \sin \phi . \qquad (*)$$ How does one generate a description of how $\phi$ depends on $\beta$? My attempt From inspection (i.e. drawing pictures) one finds that for $\beta<1$ there is a…
DanielSank
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2
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2 answers

Solving transcendental equation involving exponential functions

I'm trying to numerically solve the following transcendental equation: $$(\alpha x + \beta)=\delta e^{\gamma x}$$ with $\alpha$, $\beta$, $\gamma$ and $\delta$ real, positive constants. It is equivalent to finding the intersection between the line…
Vitality
  • 378
2
votes
1 answer

Algorithm for calculating real, positive roots of transcendental equation involving tangens

Crank ("The mathematics of diffusion", 2nd editon, 1975, p.57) describes a diffusion modelling algorithm which relies on the non-zero positive roots of $$ \tan{q_n} = -\alpha\cdot q_n$$ Typical values I use for $\alpha : 0.1, 1, 10, 15, 25, 50,...…
whoever
  • 23
2
votes
3 answers

How to compute the unknown power x in $0.15^x + 0.36^x = 1$

More generally, I have two known variables $a$ and $b$ and one unknown variable $x,$ where: $$a^x + b^x = 1, a,b \in [0, 1]$$ Is there a way to compute the value of $x$ analytically?
vampiresquid
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