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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159 views

How to solve $x e^x + x = 1$?

I have seen a mathematics-related video here, which introduce a Lambert W function to the audience: $$f(x)=xe^x\\ W(x)=f^{-1}(x)$$ Then we can use $W(x)$ to solve some transcendental equation ...
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73 views

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 $$ \tan{q_n} = -\alpha\cdot q_n$$ ...
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34 views

Uniqueness of solutions to a transcendental equation

Let $n\in\mathbb{N}$ with $n\geqslant1$ (may and should be taken large), and $\sigma>0$ be fixed. Consider the function \begin{equation} f(x):=\Big(-x^2-\sigma n^2 x + n^2\Big)+ \Big(x^2-\sigma n^2 ...
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28 views

Condition for f and g to intersect

For $f:\mathbb{R}\to\mathbb{R}$ and $g:\mathbb{R}\to\mathbb{R}$. Neither is convergent. $f(0) < g(0)$ but $f'(x)>g'(x)$ for all $x>x_0$. Is it true that there always exists an $a$ where $a>...
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3answers
92 views

Approximate solution to a transcendental equation

I'm working on a physics problem and stumbled upon the following equation: $$h=\frac{2n\pi+\arctan\left(\frac{c}{b}\right)}{b}$$ where $n \in \mathbb{Z}$, $c \in [0,20]$ and $h \in \mathbb{R}^+$. This ...
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How to derive this formula of Lambert W function?

Lately I read this in some site about a closed-form representation of Lambert W function (all branch-cuts): $$\ln\bigg(\frac{W_k(z)-1}{(\ln(z)-1+2k\pi{i})}\bigg)=\frac{i}{2\pi}\int_0^\infty{\ln\bigg({\...
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2answers
68 views

Numerical solution for transcendental equation in two variables

I have this set of two equations \begin{eqnarray} e^{-35 (y-x)} & = & \frac{x}{y} \tag{1} \\ \frac{e^{-90 x}-e^{-90 y}}{e^{-142 x}-e^{-142 y}} & = & \frac{1892}{1007} \tag{2} \end{...
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Can we say anything about the distance between sequential solutions of $\tan(k l ) = l$ to approximate $f(x) = \sum_l c_l \phi_l(x)$?

I have a differential equation with some boundary conditions whose solution is $$f(x) = \sum_l c_l \phi_l(x).$$ Here the $c_l$ are some constants which depend on $l$ and the $\phi_l(x)$ are some ...
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50 views

FInding real roots of a polynomial and transcendental equation

Show that if $$Z = 2(-H^\frac{1}{2})\cos(\theta),$$ where $Z$ is a root of the cubic equation $$Z^3 + 3HZ+G=0,$$ then $$\cos(3\theta) = \sqrt{-G^2/4H^3}.$$ Deduce that if the $H < 0$, and $G^2+4H^3 ...
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82 views

Solving a transcendental equation using matlab

How to solve x = sin(ax+b), when 'a' and 'b' takes all the real values in [0,1]? Can I use matlab to solve it? I have understood to solve the equation when 'a' and 'b' both are fixed constants. But, I ...
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1answer
128 views

Approximately solving the transcendental equation $\tan(N z)=-i\sin(z)$

everyone! I want to find a simple analytical formula for the solutions of the following transcendental equation: $$ \tan(N z) = -i\sin(z), $$ where $N>1$ is an integer, $z$ is the complex variable ...
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2answers
131 views

Is there a method for solving equations like $\sin x=x\cos(x)$ in closed form?

Is there a method for solving equations like $\sin x=x\cos(x)$ in closed form? I was looking into involute curves and ran into two equations that I'd like to find closed form solutions to: $\sin(t)=t\...
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1answer
55 views

Solving an equation with the Lambert W function

I am trying to solve the following equation \begin{equation} -e^{-i2k\ell}=\frac{k-1}{k+1} \end{equation} for $k$. I thought it might be an idea to use the Lambert W function to do so, but my ...
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35 views

Roots of transcendental equation involving linear combination of exponentials

I am looking for the solution of the following equation: $$\frac{N(x)}{D(x)} = \frac{1-A + Ae^{-\alpha x^a}}{1-B + Be^{-\beta x^b}} = r$$ all parameters are real $A,B\in [0, 1]$ $\alpha,\beta,r > ...
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53 views

Solving a system of Transcendental equations

I am trying to find the maximum value of $\lambda$, where 0 $\le$ $\lambda$ $\le$ 90$^\circ$ In the following system of equations $\lambda$, E, $\phi$ are my variables and rest all are arbitrary ...
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1answer
44 views

Finding the transcendental equation and the form of the eigenfunctions given a regular Sturm-Liouville problem

Consider the Sturm-Liouville problem: where h > 0. I need to determine the transcendental equation for the eigenvalues, as well as the "form" of the eigenfunctions. I tried finding the ...
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How do you plot the graph of the function $y^2 = \frac{(1+e^{-y})}{ x}$ .I have the graph for the problem but need to know how to obtain it manually. [closed]

The problem here is not to show the graph but if or how (i.e if it can be plotted) can I obtain it roughly without the use of a graph generator . Any help will be appreciated .
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Does a series solution to a transcendental equation always converge or can it diverge?

Suppose an arbitrary transcendental equation like $$\phi(u,z) = 0 \quad .$$ To find "analytically" the solution $u=u(z)$ that satisfies this I consider an expansion around small $z$ and ...
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72 views

Solve trigonometric equation $2x=\sin(2x)+\pi/2$

I would appreciate help in solving this equation: $$2x =\sin 2x + \frac{\pi}{2}$$ I am aware that instead of $2x$ in $\sin(2x)$ I could put the whole right part of the equation, and then again and ...
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2answers
78 views

Approximate solutions to a transcendental equation of two variables

Say I have an equation of the form: $$ y = A + B\sin (x) +C\sin (x+Dy) $$ on the domain $0<x<2\pi$. I want to get $y$ as a function solely of $x$, i.e. remove the $y$-dependence from the R.H.S. ...
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1answer
54 views

What is a transcedental Equation?

I was studying equations, and came across the term Transcedental being use for equations of the form, $$ \sin(x) -e^{x} + x^{2 }= 0 $$ From what I understand, the equations involving terms like exp, ...
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60 views

Solving the equation $-e^{-i2k\ell}=\frac{k-1}{k+1}$ for $k$

I am trying to solve the following equation for $k$: \begin{equation*} -e^{-i2k\ell}=\frac{k-1}{k+1}\, , \end{equation*} where $\ell$ is a positive number (constant). I would like to know how the ...
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1answer
62 views

General solution of $a^x = b^x\cdot(1-c) - c$

I am doing some simple models to estimate the cost of having unrealized taxed investments, but I keep running into equations that have the form of: $$a^x = b^x\cdot(1-c) - c$$ Or even, $$a^x = b^x\...
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3answers
115 views

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?
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1answer
62 views

Sturm Liouville Problem Transcendental Equation

I am trying to solve the following problem $$ X''(x)+ \lambda X(x)=0$$ $$ X'(0)+2X(0)=0$$ $$ X'(1)=0$$ Show that $$ \tan\left( \sqrt{ \lambda } \right)= -2/ \sqrt{ \lambda } $$ With the ...
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1answer
77 views

Solve transcendental equation: At $α\ll 1$, $\left|\cos x + \alpha \frac{\sin x}{x}\right| > 1$, determine the width of of the $k$-th zone at $k\gg1$.

To solve this transcendental equations approximately: At $\alpha \ll 1$ find the positive solution of inequality: $\left|\cos x + \alpha \frac{\sin x}{x}\right| > 1$, they are divided into series ...
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86 views

Solve transcendental equation : $\tanh(\alpha x) =\arctan(x)$, at $0<\alpha-1 \ll 1$ and at $\alpha\gg 1$?

To solve this transcendental equations approximately : Preivous: $\tanh(\alpha x) =\arctan(x)$, at $0<\alpha-1 \leq 1$ and at $\alpha\geq 1$. Edit: $\tanh(\alpha x) =\arctan(x)$, at $0<\alpha-1 ...
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5answers
69 views

how do you find the $x$ value for $-\sin x+\cos x=0$

Find the sationary points of the curve and their nature for the equation $y=e^x\cos x$ for $0\le x\le\pi/2$. I derived it and got $e^x(-\sin x+\cos x)=0$. $e^x$ has no solution but I don't know how ...
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54 views

Finding intersection of two curves for an area-between-curves problem

The question is to find the area enclosed in the first quadrant bounded by the line $y=\ln x$ , the line $x=2$, the curve $y=\frac{1}{x^2}$ and the x-axis. I have drawn a rough sketch. I am unable to ...
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2answers
117 views

Please help to find out the general solution of the following equation : ${ (x^2 - 7x + 11)}^{x^2 - 13x +42}=1$

For this equation : ${ (x^2 - 7x + 11)}^{x^2 - 13x +42}=1$ The integer solutions of $x$ found by WolframAlpha using inverse (logarithmic) function are $ 2 , 5 , 6 , 7 .$ Why it cannot find the other ...
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1answer
69 views

solution of $x^x = ax^b$

I am interested in the solution $x > 1$ of the transcendental equation $$x^x = ax^b$$ with $a > 1$, $b \in \mathbb{R}$. I am looking for both analytical and numerical methods, as well as ...
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1answer
255 views

Is it hopeless to try and solve this equation analytically?

Can this equation be solved with analytical methods, or is it only numeric methods since current mathematical tools don't go that far? Its complex roots are the same as the roots of the zeta function ...
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1answer
68 views

Use Lambert W Function to Solve $\frac\alpha\beta\ln(x)-x+c=0$

I need to solve this equation for x: $\frac{α}{β}\ln(x)-x+c=0$ Apparently it has to be done with the Lambert W function. I think the answer is $x=\frac{α\operatorname{W}(-\frac{βe^-\frac{Cβ}{α}}{α})}{...
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55 views

How many solutions does this equation have for different values of l?

I am trying to solve the following equation for $k$: \begin{equation} e^{2i\cdot l\cdot k}=\frac{k-1}{k+1} \, , \qquad l\in \mathbb{R} \end{equation} I already know that the equation restricts the ...
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1answer
130 views

Solve: $P_n^m(\cos\theta_w)=0$ for $\theta_w$. Zeros of Associated Legendre Polynomial

I am working on boundary value problems with the associated Legendre Polynomial and have the condition that: $$\frac{P_n^m(\cos\theta_w)}{\sin(\theta_w)}=0$$ I am trying to solve this equation for $\...
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1answer
148 views

Finding approximate analytic solution of transcendental function, possible Lambert function?

My original problem is $$g(r)=\ln(1+r)-a\sqrt{1-\frac{1}{(1+r)^2}}$$ where $r\geq0$ is a variable, $a\geq0$ is a constant. And I'm trying to find the root where $g(r)=0$. This is an transcendental ...
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2answers
58 views

If $y=\sin^2(a+\delta)$, then is there an expression for $\sin^2\delta$ in terms of $a$ and $y$?

Consider this equation: $$y = \sin^2(a+\delta)$$ Can I get the expression of $\sin^2 \delta$ from this by any trigonometric manipulations? or is this a transcendental equation and can only be solved ...
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176 views

Solution to an equation with logarithms of type $x\log(x) + ax + b = 0$

I've encountered a transcendental equation with logarithms like $x\log(x) + ax + b = 0$, and I'm wondering if there is a closed-form solution for the equation.
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3answers
124 views

How to solve $5^x-4^x-3^x-2^x-26=0$ by hand?

Is there a way to solve $5^x-4^x-3^x-2^x-26=0$ by hand? Added for clarity: I can test values and quickly find $x=3$ is a solution and can show that it is the only one. What I am curious about is if ...
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2answers
573 views

How to solve $\log_2(x)+\log_{10}(x-7)=3$ using high-school math?

A question given in a grade 12 "advanced functions" class, asks to solve $\log_2(x)+\log_{10}(x-7)=3$ with a hint to change bases. The given hint suggests the base of the second logarithm is ...
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123 views

Why do Fresnel-integrals contain $\sqrt{\pi}$?

The non-elementary functions $$ F(x) = \int \sin(x^2)\mathrm dx $$ $$ G(x) = \int \cos(x^2)\mathrm dx $$ will yield $$ F(x) =\sum_{k=1}^{\infty} (-1)^k \frac{x^{(4k+3)}}{(2k+1)!(4k+3)}$$ $$ G(x) =\...
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1answer
135 views

Solve $(x^{2}+1)e^{-x}=u$ for $x$?

I need to solve the following equation for $x$ $(x^{2}+1)e^{-x}=u$ I tried Lambert's W function but couldn't find solution for $x$.
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1answer
101 views

How do I find the intersection of two opposing involutes?

I have two involutes of a circle. They are drawn from the same base circle, but one is offset from the other and faces the opposite direction (see plot below). The parametric equations for the first ...
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27 views

Reducing transcendental equation

How would one solve or reduce an equation like this: $\frac{exp(-(-v_{11}+x_1)^2-(-v_{12}+x_2)^2)}{\sum_i^N exp(-(-v_{i1}+x_1)^2-(-v_{i2}+x_2)^2)}=\frac{exp(-(-v_{11}+y_1)^2-(-v_{12}+y_2)^2)}{\sum_i^...
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1answer
92 views

Roots of transcendental equations in physics problem

I have a physics problem where I have to find for which angle and time is the target at distance $r$ and height $0$ hit if initial velocity is $v_0$. I am aware that this problem is very basic and I ...
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2answers
197 views

Approximate solution to a transcendental equation in the limit of a variable

I have the following transcendental equation: $$2\cot{x}=\frac{kx}{hL}-\frac{hL}{kx}\tag1$$ I would like to inquire whether an approximate solution to $(1)$ can be developed in the limit $h\...
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1answer
117 views

Integral $ \int_{-\infty}^{+\infty} e^{-t^2}\ln(1+e^t) \, dt $ [closed]

$$ \int_{-\infty}^{+\infty} e^{-t^2}\ln(1+e^t) \, dt $$ Hello everyone,i would like to know the result of the above integral and how to calculate or estimate it. background and progress so far:(1)...
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1answer
79 views

Transcendental Equation with Quadratic Part (Can it be solved via Lambert W function)?

There I hope to minimize an optimization problem: $$ \min_{x \in \mathbb{R}_+} f(x) = x^2 -ye^{-x^2} + r(x -d )^2,$$ where $y, d \in \mathbb{R}$ and $r \in \mathbb{R}_+$. For this equation, the most ...
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0answers
69 views

Proof that pitchfork bifurcation happens with change in parameter

This problem is a subpart of question 8.1.14 in book Nonlinear Dynamics by Strogatz. Here we have 2 differential equations: $\dot{x_1}=-x_1+F(I-bx_2)$ $\dot{x_2}=-x_2+F(I-bx_1)$ where $F(x)=\frac{1}...
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1answer
41 views

How to solve this logarithmic equation $\ln(x/C_1)+C_2/x=C_1/C_2$

I have this equation which represents two intersection points of a graph with a line. Can anyone help me please to solve it for $x$? $C_1$, $C_2$ are constants. $x$ is a variable. The first solution ...

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