Why is my derivation of the catenary wrong?

[phantomvommand]

TL;DR

I have "derived" a differential equation for the catenary, and have attached my working. It looks slightly different from the correct expression, which can be found here: https://www.math24.net/equation-catenary

Please do tell me where I made a mistake. Thank you!

Important note: I only derived the differential equation, I did not solve it.

What I think caused the mistake:
- the tangent approximation (tan(theta+dtheta) ~ tan theta + d theta

 

[BvU]

Hi,

What I think

That's what I think too
The proper way to differentiate $\tan\theta$ is not ${d\tan\theta\over d\theta} = 1$ but $${d\tan\theta\over d\theta} = {d\over d\theta}\Biggl ( {\sin\theta\over\cos\theta}\Biggr ) =\ ... $$

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