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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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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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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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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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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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 ...
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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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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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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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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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 ...
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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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 ...
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 ...