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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.

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solve $\cos(x)\cosh(x)-1=0$

I'm trying to find the limit value of this for large values of $x$, in terms of a closed form formula. However when I try to plot this using different representations I get different curves. For $\...
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2answers
99 views

$e^x+x=0$ has countable infinite many solutions

I have shown that the following transcendental equation has only one real root. But I am looking for the argument to show that it has infinitely many numbers of roots in the complex plane. Moreover, ...
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0answers
38 views

Closed form solutions to a Gaussian equation

Let $\phi(t) := \frac{1}{\sqrt{2\pi}}\exp\{-t^2/2\}$ be the standard Gaussian pdf function and $\Phi(t) := \int_{-\infty}^t \phi(u)du$ be the Gaussian CDF function. Consider equation $$ \Phi(x) + \phi(...
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2answers
40 views

Solution for $ y(x) - ae^{y(x)} = f(x) $ involving Lambert W function

I need to solve an equation of the type: $$y(x) -ae^{y(x)} = f(x)$$ with $a>0$. Furthermore, the expression for $f(x)$ can't be evaluated analytically (it's the solution of a differential ...
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0answers
34 views

Solve ${W_{-1}}'(-1/x)=-1$

This problem arose when I tried to find the maximum of $f(x)=\ln(x\ln(x\ln(x\cdots)))-x$. This can be written as $f(x)=\exp(-W_{-1}(-1/x))/x -x$ by substituting the recursion into $f$. The negative ...
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1answer
18 views

A good approximation for $\arctan(u \tan (v x))$

I am trying to find an approximate solution for the pole of a transfer function that I need to analyze in electronics. I could simplify greatly my problem untill the point where I have to solve a ...
2
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2answers
36 views

Special solution to $a+e^a\ln x = x+e^a\ln a = a+e^x\ln a$

I was messing around with the equations of the form $a+e^b\ln c$. I set two variables equal and graphed them and I noticed something that interested me enough to ask about. Let $x\in\mathbb{R}$. For ...
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1answer
42 views

A solution to a transcendental equation

I have a transcendental equation as follows: $$ e^{ax} = b+ax $$ for some constants $a>0$ and $b>0$. Is there a known solution for $x$? I tried on Wolfram and it gave a solution which ...
2
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1answer
45 views

Prove that $e^{\sin x}-\sin e^x\neq0$ $\forall x\in [-1, \pi]$, an equation in the form $f(g(x))=g(f(x))$

As the title say, we need to prove that: $e^{\sin x}-\sin e^x\neq0$ $\forall x\in [-1, \pi]$ Now, I am not sure if this can lead somewhere but we can notice that $e^{\sin x}-\sin e^x=0$ is ...
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0answers
24 views

Existence of unique root of mixed linear/logarithmic equations

I have functions $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$ and $g$ that look like this: \begin{equation} \begin{aligned} f(x) &= Ax + b + g(x) \\ g(x_i) &= -c \log(1/x_i - 1), \quad x_i\in(0,...
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0answers
54 views

An happy coincidence for the approximate solution of $x \tan(x)=k $?

Thinking more about this question where I proposed some approximate solution of the first positive root of equation $\color{blue}{x\tan(x)=k}$ for any $k >0$, I notice that, for the $[3,4]$ Padé ...
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0answers
25 views

Express solutions in terms of $\phi$

Previously, I solved the special transcendental equation $x=e^{t/\ln(x)}$. The solution is: $x=e^{-\sqrt{t}}$ for $0<x<1.$ One can define an equation: $e^{s/\ln(x)}=e^{t/\ln(1-x)},$ for $s,t \...
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1answer
64 views

Solving for $\lambda$ in $\frac{k\lambda}{\alpha} \,\tan(\lambda R) -1 = 0$

How to analytically solve for $\lambda$ in the following equation? $$ \frac{k\lambda}{\alpha}\,\tan(\lambda R) -1 = 0. $$ where $k$, $\alpha$, $R$ are positive constants, and $\lambda$ is a ...
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0answers
108 views

How to solve equations with both logarithms and square roots, like this: $ax+b\log(x)+c\sqrt{x}+d=0$

I have an equation that looks like this: $$ax+b \log(x)+c\sqrt x+d=0$$ I know that an equation without the $\sqrt x$ can be solved using the Lambert's W function (How to solve equations with ...
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2answers
84 views

Number of solution of the equation $2\tan^{-1}|x|\cdot\ln|x|=1$ is

This questions can be solved by sketching graphs of both $1/\ln|x|$ and $\tan^{-1} |x|$. Can this be solved by algebra?
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1answer
71 views

How to solve $a^x + b^x = c \exp(d/e) - f$ for $x$

Is it possible to solve the following equation for $x$? $$a^x + b^x = c \exp(\frac{d}{e}) - f$$ Here, $a, b, c, d, e, f$ is real values and $a, b, e \neq 0$.
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3answers
69 views

exponential equations with expansion…

Solve for $x$ in the equation algebraically $$ 2^x=2x. $$ The solutions are $x =\{1,2\}.$ I have solved it but no one has validated my method. So I thought this website can help. I converted ...
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2answers
41 views

Derivatives Involving Parentheses and Trig Functions

$y=(\cos3)x^{2}$ If we were to take the first derivative of the equation, wouldn't we apply the product rule so that: $y^{'}=f(x)g^{'}(x)+g(x)f^{'}(x)$, where $f(x)=\cos3 $ , and $g(x)=x^2$ ...
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1answer
85 views

How to solve $x^x=a$ and related equations? [duplicate]

How can I solve the equation for $x$ when $x^x=2$ or any other constant? And is solving $x^{x^x}=a$ or $x^{x^{x^x}}=a$ or equations such as these even possible? What are these equations even called? ...
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0answers
70 views

Transcendental Equation. [duplicate]

I was able to solve $ \ln x = - x $ using Lambert's function. I was wondering how does one solve $ \ln x = x$ Does the solution even exist for this equation?
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3answers
95 views

How do I solve $2^x + 3^x = 5$ type exponential equations? [closed]

I was trying to solve a variety of exponential equations and I came across such equations and I am unable to solve it by the regular "taking log" method, so how do I solve such equations? (Well I know ...
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0answers
62 views

Analytical solution to $f(x)=e^{g(x)}$

Is there an analytical way to solve the equation $f(x)=e^{g(x)}$ when $f(x)=\frac{q-xe_1}{q-xe_2}$ and $g(x)=tx$, for given coefficients $q,t,e_1,e_2\in\mathbb{R}$, with $e_1<0<e_2$?
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1answer
55 views

Issue solving an equation

I'm at lost trying to solve the following equation : $$B\cdot x^{\frac{2}{3}}+C\cdot x^{\frac{1}{2}}=D$$ My research lead me to think that it's a transcendental equation but I don't know how to solve ...
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0answers
34 views

Equivalence between $\phi (z,s,a)$ and a sum of single impulses

A summed set of (negative) single impulses is given by $-\sum_{R=0}^m \frac{\sin \pi (x-(R+1))}{\pi (x-(R+1))}$ Mathematica simplifies this to a function involving the Lerch Transcendent $\phi (z,s,...
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0answers
38 views

Solution for a transcendental equation that involve logarithms

Consider the following equation $$\frac{k r}{x+r}=\frac{x \log(x)}{a-\log(x)}$$ where all the values are real and positive. Is there anyway, method or approach that will allow me to obtain an ...
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0answers
52 views

Solving a transcendental equation using the Lagrange inversion theorem

Consider the following equation: $$x=1+\frac{x}{r}e^{\frac{a x}{r(x-1)+x \kappa}}$$ where $x>0$, and $a,r,\kappa>0$ as well. Can the equation be solved for $x$ using the Lagrange inversion ...
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0answers
124 views

Generalizing the Lambert W function

Can an equation of the form $ y=e^{\frac{p(y)}{q(y)}}$ be solved for $y$ in terms of the Lambert W function? $p(x)$ and $q(x)$ are polynomials, for example $y=e^{\frac{b(y+1)}{y^2+y+\kappa}}.$
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3answers
166 views

Number of roots of the equation $x\sin x-1=0$ for $x\in [0,2\pi]$

Number of roots of the equation $x \sin x-1=0$ for $x\in [0,2\pi]$ My attempt is using the bisection method where I initially took $a=0$, $b=\frac{\pi}{2}$ as $f(a)\cdot f(c)<0$ We can ...
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5answers
650 views

Solving exponential equation by hand?

I was wondering how to solve the following exponential equation for a by hand: $$-e^{-a/2}-\frac{1}{2}ae^{-a/2} = -0.95$$ I'm able to find a solution with computer software, but am stuck when it ...
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1answer
67 views

How to solve equations like these?

If $f(x) = x^{2} - \frac{\cos x}{2}$ and $g(x) = x\frac{\sin x}{2}$. Find the points at which $f(x) = g(x)$. I'm stumped and have no idea how to proceed or solve questions like these.
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1answer
44 views

Uniqueness of solution for transcedental equation on the open set

What is the best way to prove that x=a is the unique solution for the equation $$\frac{2a}{x} = \exp (2-\frac{2x}{a}) +1$$ for $x>0$ ? Intermediate Value Theorem does not work since the interval is ...
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1answer
100 views

Transcendental equation with Bessel function

I am trying to solve the following equation for days (not a mathematician): $\sin \phi \;J_n (a \phi) = e^{in\pi} \;\sin \phi' \;J_n (a \phi')$ I need to find such $\phi'$ that the equality is ...
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1answer
125 views

Solve $a^x = 1-x$

This has the obvious solution $x=0$ but it also has a positive solution (which must be $x<1$, ignoring complex numbers) when $a>0$ and $<\frac{1}{e}$. I can solve it by fixed-point iteration ...
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1answer
81 views

Transcendental equation $x - c \sin(x)=0$ [closed]

Hy, well I have a problem with the transcendental equation $x - c \sin(x) = 0$, where $c$ is some positive constant. I tried using Newton's method for finding the roots but it didn't work well. The ...
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2answers
90 views

A complicated equation system

Are you up for a challange? Nine years ago, I experimented with some variables and came up with an equation system. I figured out there were solutions to it and worked hard to find generalized ways to ...
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1answer
113 views

How to solve exponential equation?

I want to mention that it is not my homework, just want to solve for fun. I appreciate any hint how to solve it. The exponential equation is given: $2^x + 3^x = 10000$ My initial thought was to use ...
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3answers
239 views

Solve a transcendental equation [closed]

I have the following complicated equation $e^{a t} = 2 \cos(bt)$, needs to be solved for t. Any possibility of getting analytic solution? If not, can we solve it in Mathematica?
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0answers
69 views

Closed-form solution for a transcendental-equation with specific coefficients?

I would like to know whether there is a closed-form solution in $x$ for the equation: \begin{align*} &e^{-(\alpha\delta+(\gamma-\phi)\frac{\alpha}{\alpha - 1})x} -\frac{\alpha(r+\delta+\phi-\...
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2answers
145 views

Solving transcendental equation involving exponential function

I want to find analytical solutions of the equation $$(αx+β)=δe^{γx}, \qquad (1)$$ which is exactly this one: Solving transcendental equation involving exponential functions The substitution $y = -\...
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2answers
289 views

How to solve $e^{2x - 1} = -x^2 + 3x - 0.25$

I know the answer is $x=0.5$, but I only know it because my calculator showed me the intersection point of the two graphs. What I want to know is how to get to this result algebraically. I've tried to ...
2
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0answers
59 views

Explicit solution(s) for equations of the form $c=x^{ab}(d+ex^{-a})$?

Is it possible to find a solution in $x$ for equations of the form $c=x^{ab}(d+ex^{-a})$? Of course, if we look for real solutions, the answer will depends on the values of a,b,c,d and e. But is ...
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0answers
53 views

Transcendental equatiom, multiple roots

Suppose the following transcendental equation $$ F(u)=\sin u(A-4u^{2})+u(B+Cu)-D=0, $$ where $A=\pi^{2}(x^{2}+y^{2})$, $B=2\pi(1-x^{2}-y^{2})$, $C=4y$, $D=\pi^{2}y$, are the constant values, and $x\in[...
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2answers
163 views

Solve for $x$: $10^x=x+100$

I don't know how to solve it: $$10^x=x+100$$ The original question was how many answers are there, so I drew the functions, then I see there are 2 answers (cuts?). I really want to know how to solve ...
2
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0answers
56 views

Inverse function and transcendental equation

Suppose the parallel arc of the radius $r$, latitude $\varphi$, centered at $[0,q]$ passing through three points $[0,2\varphi/\pi]$, $[\cos\varphi,\sin\varphi]$, and $[-\cos\varphi,\sin\varphi]$, $$ ...
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1answer
46 views

How to find the first root of a function with 1 variable?

I have a function with 1 variable x (positive number), I want to find the smallest root (the smallest x, where f(x)=0), I know that my root is bigger than 6.5. and I have graphed the function for ...
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0answers
175 views

Find the roots to the following transcendental equation?

How can I most easily find the roots to the following transcendental equation? $$ f(K)=K \cot(K L)+\sqrt{2V-K^2}\coth\Big( \sqrt{2V-k^2}\frac{a}{2} \Big) \tag{1} $$ Where $L,V$ and $a$ are constants....
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1answer
114 views

Golden angle for a circle: another definition

The accepted definition for the golden angle is based on the ratio of circle arcs, i.e. if $\alpha_0$ is the golden angle in radians, then: $$\frac{a}{b}=\frac{b}{a+b}$$ Where $a=\alpha_0 R$ and $b=(...
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2answers
48 views

How to show the relation of the Inverse trigonometric functions

I want to show the relation; $$ \arccos[-1+\frac{Q^2}{2}]=2\arccos[\frac{Q}{2}]$$ Q is a real number. I used Mathematica 8 and I checked this relation is correct. But I don't know how to prove it. ...
2
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0answers
181 views

Simultaneous transcendental equations involving digamma functions.

Is there a way (can I hope) to express the solutions of the following coupled transcendental equations in a closed form (in terms of Lambert, hypergeometric or other special functions) : \begin{...
6
votes
0answers
198 views

I am a physicist with a difficult equation (quadratic exponential) I am curious about. No luck with a lit review, sympy or Mathematica.

I have an equation I have been playing with (the variable is $x$ and the constants are positive and real): $$( r_1 - x )^2 \frac{ a r_3 e^{x / r_2} + b r_2 e^{x / r_3} }{\left( a e^{ x / r_2} +...