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.

learn more… | top users | synonyms

0
votes
1answer
32 views

Finding the number of roots of the equation [duplicate]

The equation $x^{13}-e^{-x}+x-\sin{x}=0$ has No real root More than two real roots. Exactly two real roots. Exactly one real root. I tried doing with the odd derivative and check whether the ...
1
vote
0answers
19 views

A problem on Newton-Raphson method

The function $f(x)=0$ has a simple root in the interval $(1,2)$. The function $f(x)$ is such that $|f(x)|>3$ and $|f''(x)|\leq 4$ for all $x\in (1,2)$. Assuming that the Newton-Raphson method ...
-4
votes
1answer
74 views

Find the values of x with proper solution [duplicate]

Solve the equation:(without the help of a computer). Solve it by hand or manually with explanation. $$ \large x^2=2^x$$
0
votes
1answer
36 views

How to find the roots of a equation involving log terms?

This question was in my test and I am not sure what to do with it let $f:(0,\infty)\rightarrow \mathbb{R}$ be given by $$f(x)=\log x-x+2$$ then its number of roots of $f$are. So putting it ...
0
votes
1answer
120 views

Exact solution to system of first order ODEs

I am trying to solve a system of 1st order ODE's. $m(r)$ and $P(r)$ are real functions of $r$ and should be positive. $a$ and $b$ are just constants. $$\frac{d m(r)}{dr}=\pi(\frac{P(r)}{a}+b)r^2\\$$ ...
1
vote
1answer
53 views

How to know if I can't solve an equation with “standard” methods?

I'm particularly fascinated by transcendental equations whose posses closed form solutions and when I pose some of them to my friends or teachers I heard a lot of "You can't solve this in closed form" ...
4
votes
3answers
91 views

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
1
vote
0answers
24 views

Self-consistency transcendental equation for Curie-Weiss model

In physics, the ferromagnetic Curie-Weiss spin model leads to a transcendental self-consistency equation for the magnetization $m$ of the form $$ m = \tanh(J m +h)\ , $$ with $J>0$ and ...
0
votes
2answers
36 views

To find roots of given equation

If $f(x)=x^2$ and $g(x)=x\sin(x)+\cos(x)$ then i have to find number of points$(x)$ such that $f(x)=g(x)$ I write $h(x)=f(x)-g(x)$. So $h(x)=x^2-x\sin(x)-\cos(x)$ $h'(x)=x(2-\cos(x))$. Since $2-\cos ...
0
votes
2answers
48 views

To find number of real solutions of equation $\left(\frac {9}{10}\right)^x =-3+x-x^2 $

To find real solutions of $\left(\frac{9}{10}\right)^x = -3 + x -x^2$ I differentiate it to get $\left(\frac {9}{10}\right)^x log(\frac{9}{10})-1 + 2x=0 $ As x goes to $+\infty$ this goes to ...
2
votes
0answers
49 views

Solving for a $v$ in $\sum a_i e^{b_i (z^2+d_i) + c_i v}$

I have an equation in complex domain, $$P(e^u,e^v)=\sum_{i=1}^{N} a_i e^{b_i u + c_i v}=0 \;\;\;\text{(A)}$$ and by redefining, at the roots (I'm only showing work for one root), the first ...
2
votes
2answers
54 views

For what values of $x$ is $\cos x$ transcendental?

For what values of $x$ is $\cos x$ transcendental? Is there any way I can figure out the values of $x$ where $\cos x$ is transcendental or do I have to check individually for every $x$ whether it is ...
0
votes
1answer
51 views

Difficulty in solving transcendental equation

Let $A,B,C$, and $D$ be positive constants. What's the most concise way to express $x$ in the equation below? $$ A = B\arctan(x/C)+Dx,$$ where $0<x<1$ and we know that $C=\cos(30^\circ)$.
2
votes
0answers
35 views

Find the positive root of the equation $\cosh x+\cos x-3=0$, other than numerically

I know you are able to find the root of the equation by using Newton-Raphson method. But is there any other way? $$\cosh x+\cos x-3=0$$ I thought maybe you could say that $-1\leq \cos x \leq 1$. So ...
1
vote
0answers
36 views

Existence of solution for equation with erfc

I'm looking for the self-consistent (e.g. input needs to be the same as output) solution of $r$ in $$ r = \frac{1}{2}\operatorname{erfc}(z(r)) $$ where $\operatorname{erfc}$ is complimentary error ...
2
votes
1answer
60 views

Table Maker's Dilemma

I am trying to understand the TMD based on this doc. As per my understanding, For transcendental functions i..e (log2 log10 1oge, exp,sin, cos,tan...) the exact value for an input cannot be ...
0
votes
0answers
58 views

Derivative of Inverse Mills Ratio (Conditional expectation of normal distrbution is strictly increasing)

I'm trying to show that the derivative of the inverse Mills Ratio is bounded between zero and one. Essentially, for a standard normal distribution, I want to show that ...
1
vote
1answer
36 views

How can I solve $\beta^2=\frac{m^2g}{h}\left(-\frac{\beta t}{m}+e^{\frac{\beta t}{m}}-1\right)$ for $\beta$?

This equation arose when I tried to find out how to derive $\beta$ in Stokes' Drag Force $F=\beta v$ as a function of the time $t$ it takes a mass $m$ to hit the ground after falling from a height ...
1
vote
2answers
151 views

How to solve an equation with a tangent divided by a logarithm?

Here is an equation and I've never met this kind before. I would greatly appreciate your help. Maybe it's ridiculously simple and I overlook something? $$-12=\frac{\tan(x+4)}{\log(x+0.25)}$$
1
vote
1answer
24 views

Two Poisson r.vs with different rates may have same value at some argument

I am curious whether two Poisson distributions with two different rate parameters may have the same probability value at some positive integer argument i-e I am trying to solve $$\frac{e^{-\lambda ...
4
votes
2answers
213 views

Exponential and ln function

It seems quite simple but how would I find the exact solution for: $$\exp(x) = -\ln(x) $$ I'm not too sure where to start?
2
votes
1answer
38 views

When is $a(z) = b(c(z)) $?

Let $a(z)$ be a given transcendental entire function. When is $a(z)=b(c(z))$ where $b,c$ are also transcendental entire functions ? How to find such $b,c$ ? In particular when $a$ is given by a ...
1
vote
0answers
57 views

solve $a\cdot e^{b\cdot x}+c\cdot ln(x)=0$

Is it possible to find the analytical solution of $a\cdot e^{b\cdot x}+c\cdot ln(x)=0$? Is that a transcendental equation?
2
votes
0answers
62 views

Solving $-1=e^a-2e^{av}$ as part of a equation system

Problem Given $f_2(x)=e^{ax-b}+c$ with $x \in \left(0,1\right)$, I am trying to calculate the parameters $a,b,c$ in respect to the following constraints: $$ \begin{align} f_2(0) &= 0 \\ ...
1
vote
0answers
65 views

What families of transcendental equations do we have solved?

I'm particularly interested in transcendental equations but searching in internet gives me only results about the classical linear-exponential equation (which is solved with Lambert's W) and its ...
-1
votes
1answer
76 views

Transcendental equation $2 x n\cot (2x)= x^2 - n^2$

I have a transcendental equation and I have not a mathematical superiour formation (I'm an hydraulic engineer) necessary to solve it. The equation is : $2 x n\cot (2x)= x^2 - n^2$ or (same equation) ...
1
vote
1answer
37 views

How to find the Roots of the Derivative of two summed Gaussians.

Let $G$ be the Gaussian $$G(t,w,c,h)=w\cdot e^{-\dfrac{(t-c)^2}{2h^2}}$$ for some real parameter $t$ and the real constants $w$, $c$, and $h$. Now, let $F$ be a function defined in terms of $G$, ...
4
votes
3answers
92 views

Simple Logarithms Equation

$$3^x = 3 - x$$ I have to prove that only one solution exists, and then find that one solution. My approach has been the following: $$\log 3^x = \log (3 - x)$$ $$x\log 3 = \log (3 - x)$$ $$\log 3 ...
10
votes
3answers
90 views

Global Minimum of $f(a) = \int _{-\infty}^{\infty} \exp\left(-|x|^a\right)dx, a\in(0,\infty)$

Playing around with the Standard Normal distribution, $\exp\left(-x^2\right)$, I was wondering about generalizing the distribution by parameterizing the $2$ to a variable $a$. After graphing the ...
0
votes
1answer
146 views

Solve $x^a = 1 - \exp(-x)$ for $x$

I would like to obtain a closed-form solution for the equation $x^a = 1 - \exp(-x)$, in which $x$ is the (real strictly positive) unknown and $a$ is a real positive parameter. So far, I have tried ...
0
votes
1answer
91 views

Why a transcendental equation can not be analytically evaluated

I'm reading this book in Classical Mechanics and they derive an equation for the time a projectile takes to reach the ground once is fired (accounting for air resistance): ...
2
votes
0answers
47 views

Can we solve $ae^x+b=x^{-\alpha} $ using LambertW function?

In the equation, a,b,$\alpha$ are all constants. Can we solve the equation?
3
votes
0answers
36 views

It is possible to talk about the degree of a transcendental equation?

When we deal with algebraic equations involving polynomial and so on we know what the degree of the equation is and this tells us how many solutions we'll find (at least in complex numbers). But this ...
2
votes
0answers
66 views

Solving Kepler's Equation

I've been working on simulating orbits. I've found that, when solving Kepler's equation, $M = E - \varepsilon\sin{E}$, I'm unsure about the solution to use. For a true anomaly $< \pi$, using the ...
4
votes
5answers
264 views

How to solve $4^x+\sin(x)=10$

$$4^x+\sin(x)=10$$ I would use a log function to solve it but I don't know what to do with $\sin(x)$. What is the $x$ value of the exponent?
1
vote
2answers
61 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
vote
1answer
69 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
votes
1answer
46 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
103 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 ...
10
votes
3answers
232 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
votes
2answers
215 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
votes
2answers
72 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
376 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
304 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
72 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
77 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) ...
4
votes
1answer
114 views

If $\frac{x-1}{e^x-1} = y$ then $x=?$

I have following equation: $$\frac{x-1}{e^x-1} = y$$ I want to solve this equation such that I have the value of $x$ in the term of $y.$ i.e. inverse of the equation
1
vote
0answers
30 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
vote
3answers
310 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
vote
1answer
143 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. ...