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.

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-2
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2answers
45 views

How to solve this equation to find a closed-from for x? [on hold]

I want to find the value of $x$, i.e., $x=$ $$Ax+10^{-Bx}=C$$ Any suggestions?
1
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2answers
42 views

Solving $x^y = y^x$ analytically in terms of the Lambert $W$ function

I'm interested in deriving the solution for $y$ in terms of $x$ given $x^y = y^x$ using the Lambert $W$ function. Wolfram Alpha states: $$y = - \frac{x\ W\left(-\frac{\log(x)}{x}\right)}{\log(x)}$$ ...
1
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1answer
42 views

the roots & the limit of $2^{x^{\cos(x)}}\sqrt{\cos(x)}=2^{x}$

If $$2^{(2\pi)^{\cos(2\pi)}}\sqrt{\cos(2\pi)}=2^{2\pi}$$ Can you obtain or is it plausible to find the roots and the limit of $$2^{x^{\cos(x)}}\sqrt{\cos(x)}=2^{x}$$ if $0 < \cos(x)$ and $0 < ...
0
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1answer
36 views

How to calculate the shielding time and determine the time step

The problem is illustrated as follows. A shielding plate scans over a target plate at a constant speed $v_{scan}$ and dynamically shadows the target plate to adjust the exposure time of the light ...
3
votes
1answer
69 views

Solving an equation including $e^{-x}$ with the Lambert W function

Given two functions of $x$, namely $f(x)$ and $g(x)$, where $$f(x)=x^2-4x+8$$$$g(x)=3xe^{-x}$$ the shortest distance between the graphs of the functions is sought. I begin by defining a function ...
6
votes
3answers
145 views

A curious equation containing an integral $\int_0^{\pi/4}\arctan\left(\tan^x\theta\right)d\theta=\frac{\ln2\cdot\ln x}{16}$

I came across an interesting problem that I do not know how to solve: Find $x>0$ such that $$\int_0^{\pi/4}\arctan\left(\tan^x\theta\right)d\theta=\frac{\ln2\cdot\ln x}{16}.$$ Could you ...
7
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2answers
149 views

Solving $\ln{x}=\tan{x}$ with infinitely many solutions

Lets take $f(x)=\ln{x}$ and $g(x)=\tan{x}$ When $f(x)=g(x)$ that is $\ln{x}=\tan{x}$, we see that the graph is like: Hence we see that there are infinitely many solutions to $x$ but the two ...
0
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2answers
53 views

what is the best approximation for sine?

can you tell me which is the best approximation for cosine/sine functions. It should also reduce the computational complexity. I've already tried the Bhaskara-1 approximation. Can you suggest me ...
6
votes
3answers
120 views

How to solve $x^2 = e^x$

The question is to find $x$ in: \begin{equation*} x^2=e^x \end{equation*} I know Newton's method and hence could find the approx as $x\approx -0.7034674225$ from \begin{equation*} ...
5
votes
2answers
229 views

Solution of a Lambert W function

The question was : (find x) $6x=e^{2x}$ I knew Lambert W function and hence: $\Rightarrow 1=\dfrac{6x}{e^{2x}}$ $\Rightarrow \dfrac{1}{6}=xe^{-2x}$ $\Rightarrow \dfrac{-1}{3}=-2xe^{-2x}$ ...
2
votes
4answers
65 views

Always transcendental? $a + b e^{-x} - f(x) = 0$

I want to choose $f(x)$ such that the equation $$a + b e^{-x} - f(x) = 0$$ is analytically solvable. Ideally, I want $f(x)$ to be some function that is symmetric about 0 and everywhere positive, like ...
1
vote
3answers
68 views

Ugly expression. Cant tell if I can simplify. Is there a general method for simplifying basic algebraic expressions?

Let $\rho, \omega, c > 0$ and let $\alpha \in [0,1]$. I have managed to calculate \begin{align*} \frac{\rho}{\alpha x^{\alpha-1}y^{1-\alpha}}&= \eta\\ \frac{\omega}{(1-\alpha) ...
1
vote
0answers
25 views

Proof that for $\alpha > 0$ and $\tau \geq 0$ , P($s$) = $(s+\alpha+ e^{-s\tau})$ has no solution less than zero

P($s$) = $(s+\alpha+ e^{-s\tau})$ Proof that for $\alpha > 0$ and $\tau \geq 0$ the polynomial has no solution less than zero. I am having difficulty proving that for $\alpha > 0$ and $\tau ...
1
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3answers
269 views

Request for help to solve an equation with LambertW: $ (x^2-4\,x+6) e^x =y$

I want to solve the following equation: $$ (x^2-4\,x+6) e^x =y \tag{1} $$ It looks a bit like the following equation: $$ x e^x =y \tag{2} $$ Since the solution of equation (2) is: x=LambertW(y), ...
1
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1answer
105 views

Can I solve this with a Lambert Function?

New to W-Functions and do not understand it properly. How do I solve this equation? I know about numerical solutions (or graph solution), but I'm interested in pure algebraic solution if it exists. ...
1
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3answers
66 views

For what k values are there more than one solution to the following equation?

Suppose we have the equation, where k is some constant: $0 = (x+k)e^{-(x+k)^2} + (x-k)e^{-(x-k)^2} + xe^{-x^2}$ A trivial solution exists, where x = 0. How can I figure out what range of values for ...
1
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2answers
109 views

How to solve this equation for x? $0 = (x+k)e^{-(x+k)^2}+(x-k)e^{-(x-k)^2}$

How can I solve this equation: $0 = (x+k)e^{-(x+k)^2}+(x-k)e^{-(x-k)^2}$ To find x in terms of k?
2
votes
2answers
51 views

Solving $z=w/2-\sin(tw)/(2t)$ for $w$

Is it possible to solve $$z=\frac{w}{2}-\frac{\sin(tw)}{2t},$$ for $w$? My first thoughts were that we would have to be careful about the domain of $f(w)$ so that the inverse was actually a function ...
2
votes
1answer
75 views

How to solve the following equation? $\left(\sqrt{u^2-1}+u\right)^{1/u}=\pi ^{1/\pi }$

I have no clue: $$\left(\sqrt{u^2-1}+u\right)^{1/u}=\pi ^{1/\pi }$$
1
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2answers
50 views

Equations involving factorial/Gamma function

Are there any known methods to formally solve equations like: 1)$x^3!+(2x^2)!-x!+3=0$ 2)$x!=e^x$ ($0$ is trivial but there must be another one) 3)$(2x!)^2+x!-1=0$ 4)$x!!+x!=7$ I don't need ...
1
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0answers
34 views

Understanding simple transcendental field extensions

I would like to understand the definition of a simple transcendental extension and the theorem that states all such extensions are isomorphic. So for example, if $K \subseteq \mathbb{C} $ is any ...
1
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1answer
71 views

find an inverse function of complicated one

Let $f:\mathbb{R}\rightarrow \mathbb{R}$: $$f(x) = \sin (\sin (x)) +2x$$ How to calculate the inverse of this function? So far i searched a lot in the internet but i didn't find any easy algorithm ...
0
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1answer
64 views

Two kind of equations involving natural log and exponentiation

I know how to solve equations using Lambert's W function like $xe^x=k$ or $e^x+x=k$ But how can I solve this two kinds of equations involving natural log ? $e^x \ln(x)=k$ and $e^x+\ln(x)=k$ I ...
1
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0answers
39 views

Solving the transcendental equation $Li_{3}(e^{-kx}) + x\, Li_{2}(e^{-kx}) = k\, x^3$.

I need to solve the following equation: $Li_{3}(e^{-kx}) + x\, Li_{2}(e^{-kx}) = k\, x^3$ for $x\in\mathbb{R}^{\ast}$ and where $k\in\mathbb{R}^{+}$. Here $Li_{3}$ and $Li_{2}$ are the third and ...
1
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1answer
77 views

Transcendental equations involving more than 2 terms

I now how to solve transcendental equations involving only two terms like: $xe^x=k$ $x=W(k)$ Where W(x) is the Lambert's Omega function. But how can I solve (for $x$) a more general case? Like: ...
0
votes
2answers
33 views

Solving equation for x values

What methods I can use to solve for $x$? And how to do it? $$2x + ( 1 + \cot(x/2) ) / \sin(x/2) + 0.8 = 0$$
3
votes
3answers
127 views

explicit solution for transcendental equation

Does anyone knows whether there is an explicit, analytical solution for transcendental equations of the form $A x + B \tanh(C x) + \coth(x) = 0$, where $A, B$, and $C$ are positive real constants?
2
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2answers
75 views

Solution to $x^\alpha + p x = q$?

I was wondering if there was any tricks, similar in spirit to the Vieta's substitution, that would apply the equation $$ x^\alpha + p x = q, $$ where $p,q$ and $\alpha$ are real constants. In ...
0
votes
3answers
166 views

Exact values of the equation $\ln (x+1)=\frac{x}{4-x}$

I'm asking for a closed form (an exact value) of the equation solved for $x$ $$\ln (x+1)=\frac{x}{4-x}$$ $0$ is trivial but there is another solution (approximately 2,2...). I've tried with ...
0
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0answers
26 views

solving/approximating the transcendental inequality $c \le αx + β(b^x) + γx(b^x)$

I couldn't find a representation of $x$ using Lambert $W$ function and I doubt this is even possible. Assuming there is no clean solution and numerical methods must be used, is there a way to ...
0
votes
0answers
56 views

Solving transcendental equation with the unknown on both sides

I need to find the point of intersection between a line segment and a sine wave. Line: $y=-2x+1$ Wave: $y=\sin x$ I put both equations together. $-2x+1=\sin x$ I attempt to isolate $x$. $\sin ...
0
votes
0answers
37 views

Determine if the graph $f(x) = \ln(x)$ has any critical numbers

Determine if the graph $f(x) = \ln(x)$ has any critical numbers. The derivative would be $f'(x) = \frac{1}{x}$
3
votes
2answers
258 views

How to solve: $\frac{x}{\log_2(x) }= y$

For example, I can solve: $x \log_2(x) = y$ $x \log_2(x) = x \log_e(x) / \log_e(2) = e^{\log_e(x)} \log_e(x) / \log_e(2)$ $e^{\log_e(x)} \log_e(x) = y\log_e(2)$ $e^{W(z)} W(z) = z$, where W(z) is ...
1
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2answers
46 views

Find roots for an equation with quadratic, linear and log terms?

I'm wondering if there exists a closed-form or analytic expression for the roots of an equation of the form $ax^2 + bx + c\log x=0.$ considering the natural $\log$. Wolfram alpha is leading me to ...
2
votes
2answers
145 views

Solving the equation $\ln(x)=-x$

I tried solving this equation for a long time but did not succeed. Any help is appreciated. $$\ln x=-x$$ I am not sure the tag is correct, I am not familiar with English mathematical terms. Please ...
1
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1answer
54 views

An equation with multiple solutions: finding the maximum of the function of the solutions.

Possibly, this is a bad (stupid) question, but sometimes some discussion helps. I have a fixed point equation (involving $\tanh$). I would like to derive the dependency of some function of the fixed ...
0
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1answer
57 views

Why a trigonometric function doesn't satisfy a polynomial equation?

Why can't I have a trigonometric function as an input to a polynomial equation?
0
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0answers
46 views

Showing no algebraic solution exists for a given equation

Let $f(x)=g(x)$ be an equation (1) where at least one of $f$ and $g$ are transcendental functions. Let $h(x)=f(x)-g(x)$. If it can be shown that $h^{-1}(0)$ is non-algebraic, that implies that there ...
3
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1answer
73 views

Solving equation involving self-exponentiation

How do I solve the equation $\displaystyle x=ay^2(by)^{\frac 1y}$ for $y$, where $a$ and $b$ are constants? I've been trying to manipulate this into a form on which I can use the Lambert W function, ...
1
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0answers
48 views

Specialized numerical method for transcendental equation

Is there any specialized, very fast, numerical method for solving equations of a type $$ e^{-px-q} = \frac{ax^2 + bx + c}{kx + l} $$ wher all $ a, p, q $ are strictly positive? To be more precise, ...
0
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1answer
31 views

Characterizing conditions for $\tanh{(kx-b)}=x$ to have 1/2/3 fixed points.

I am trying to understand what are the conditions for $\tanh{(kx-b)}$ to have 1 or 2 or 3 fixed points. That is I am trying to characterize conditions on $k$ and $b$ for which equation ...
3
votes
2answers
138 views

How to solve Kepler's equation $M=E-\varepsilon \sin E$ for $E$?

I'm trying to create a program to solve a set of Kepler's Equation and I cannot isolate the single variable to use the expression in my program. The Kepler Equation is $$M = E - \varepsilon ...
2
votes
3answers
68 views

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 ...
4
votes
1answer
136 views

Inverse of $f(x) = xe^x-x$

I'm wondering if there is a way to obtain the inverse of the function $y=xe^x-x$. I am aware of the use of Lambert's W function in the inverse of $xe^x$ but as can be seen this is a different animal ...
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2answers
45 views

Solving equation $-t-0.2+ 0.2e^t=1$

I don't know how to solve this one, please give me some clues. $-t-0.2+ 0.2e^t=1$ $e^{-t}(0.16e^{2t}+0.48e^t+0.36)=1$
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2answers
97 views

Solving a transcendental equation

How do I go about solving the following equation? $$x = A + B \log\left( \cosh\left(\frac{x}{C}\right)\right)$$
0
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1answer
54 views

How to solve the non-linear equation $-(a+c\,e)\left(\exp(-b/(a+c\,e))-1\right)-c\,d=f$ for $c$?

I have this non linear equation: $$-(a+c\,e)\left(e^{-\frac{b}{a+c\,e}}-1\right)-c\,d=f$$ The only unknown is $c$. All the coefficients ($a$, $b$, $c$, $d$, $f$) are real non-null costants. How can I ...
1
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0answers
47 views

Find the positive root of the equation $ce^{-c}-2(1-e^{-c})^2=0$

Can you help me find a root for $c$ in the equation below? $$ce^{-c}-{10\over5}(1-e^{-c})^2=0$$ By expanding this I got, $$ce^{-c}-2 + 4 e^{-c}-2e^{-2c}=0$$ now grouping, ...
0
votes
2answers
115 views

Lambert W function with rational polynomial

During my research I ran into the following general type of equation: $$\exp(ax+b)=\frac{cx+d}{hx+f}. $$ Does anyone have an idea how to go about solving this equation? Thanks in advance.
0
votes
1answer
66 views

Number of solutions of a transcendental function

I am studying time-delay differential equations now. When I read "textbook" material, my teacher wrote it, this lemma occurs to me. For a transcendental function, ...