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I have an equation describing what geometrical parameters differentiate mechanically bistable membranes form monostable membranes. Long story short, I have simplified all of the parameters (such as distance to apex, and angular opening) and got the following single variable equation, which I cannot seem to wrap my head around:

$$\arcsin(\frac{20x}{100+x^2})\ * \sqrt{(\frac{50}{x})+(\frac{x}{2})} = 1.339 $$

(where $x$ is the height of the buckled membrane)

I managed to find the solution numerically, and $x = 0.90$ mm, however I'm really curious of how to solve it analytically.

Thanks to anyone who is willing to try to tackle this (according to me) tough equation!

UPDATE:

So after doing some searching, I found out that the solution to this equation is not in closed-form and so only Lambert W function can be used to approximate the answer. Now, what I still do not understand is why doesn't this equation have a closed-form solution? What are the criteria for an equation to have closed-form solution?

Thanks again for any help!

Maskurate
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  • Why is it not possible to solve it analytically? I'm sorry if it's something obvious that I'm missing. I'm not a mathematician, I'm a biomedical engineer and I find this problem really interesting, even though I already have the solution that I need. – Maskurate Oct 19 '23 at 05:50
  • An equation doesn't have a closed-form solution if it doesn't have a closed form solution. There is a well-defined sense in which hardly any equations have closed-form solutions, it's just that your teachers only assign questions that do have closed form solutions, so you get the mistaken impression that most, or all, equations have closed form solutions. Here are some simple-looking equations that don't have solutions in terms of the usual functions of school algebra & intro Calculus: $2^xx=17$, $x^5+x=1$, $\cos x=x$, $x+2^x=10$, $x+\sin x=9$. – Gerry Myerson Oct 20 '23 at 06:09
  • You may find https://math.stackexchange.com/questions/9199/what-does-closed-form-solution-usually-mean to be informative. – Gerry Myerson Oct 20 '23 at 06:16
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    Thank you Gerry! It's very interesting thought. I've never thought of this, but with your examples, I now realize that indeed there are a lot of real-life equations that does not have a closed-form. I wish my professor would have explained it this way. It really makes you see it from a different perspective :D – Maskurate Oct 20 '23 at 11:26

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