Articles of transcendental equations

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 k enable additional solutions for x to be found in addition to the trivial x = 0 solution? Context: If […]

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 correct me if I am wrong.

Solution of a Lambert W function

The question was : (find x) $6x=e^{2x}$ I knew Lambert W function and hence: $\Rightarrow 1=\dfrac{6x}{e^{2x}}$ $\Rightarrow \dfrac{1}{6}=xe^{-2x}$ $\Rightarrow \dfrac{-1}{3}=-2xe^{-2x}$ $\Rightarrow -2x=W(\dfrac{-1}{3})$ $\therefore x=\dfrac{-1}{2} W(\dfrac{-1}{3})$ But when i went to WolframAlpha, it showed the same result but in the graph: WolframAlpha Graph The curves intersect at a point… And hence there is a second […]

How do i solve $e^{ax}-e^{bx}=c$ for $x$?

How do i solve $e^{ax}-e^{bx}=c$ for $x$? The constants $a$, $b$ and $c$ are real numbers. It is the final form of a longer equation that I simplyfied. Edit: The actual equation I’m trying to solve applies to a radioactive generator of molybdenum-99/technetium-99m in my nuclear medicine department. We want to know what is its […]

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 or not?

all complex solutions of $z\sin(z)=1$?

A possibly easy question, Can we find all complex solutions of $z\sin(z)=1$ ? My try: Let $$\sin(z) = \frac{e^{iz} – e^{-iz}}{2i}$$ so we have $$ z\frac{e^{iz} – e^{-iz}}{2i}=1 $$ Not sure how can I continue ? Thanks EDIT : I have posted a related question here.

Solve equation $\exp(ax)+\exp(bx)=1$

The equation is $$ \exp\left(ax\right)+\exp\left(bx\right)=1, $$ where $a$ and $b$ are known real constants, $x$ is unknown. I would like to have the solution in form of relatively known special function (something like Lambert $W$ function, or generalized hyper-geometric $F$).

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 suggest how to approach it?

The positive root of the transcendental equation $\ln x-\sqrt{x-1}+1=0$

I numerically solved the transcendental equation $$\ln x-\sqrt{x-1}+1=0$$ and obtained an approximate value of its positive real root $$x \approx 14.498719188878466465738532142574796767250306535…$$ I wonder if it is possible to express the exact solution in terms of known mathematical constants and elementary or special functions (I am especially interested in those implemented in Mathematica)?

How do I solve this exponential equation? $5^{x}-4^{x}=3^{x}-2^{x}$

How do I solve this exponential equation? $$5^{x}-4^{x}=3^{x}-2^{x}$$