I am encountering an issue with Jacobi elliptic functions. Specifically, I am facing a challenge with the following integral:
$$\pm\int_{-\sqrt{\frac{2\mu}{\eta}}}^{u}\frac{\mathrm ds}{s\sqrt{\left(s+\sqrt{\frac{2\mu}{\eta}}\right)\left(\sqrt{\frac{2\mu}{\eta}}-s\right)}}=\sqrt{\frac{\eta}{2}}x$$
I know that the result is
$$u=-\sqrt{\frac{2\mu}{\eta}} \operatorname{sech}(\sqrt{\mu}x) $$
However, the challenge lies in understanding how to derive this result. Despite investing a significant amount of time, I am struggling to comprehend the process leading to this outcome.
Any assistance or guidance on this matter would be greatly appreciated. Thank you.
Let $$F(u,v)=\int_{-v}^u\frac{\mathrm ds}{s\sqrt{(v-s)(v+s)}}$$ Your integral in question is $F\big(u(x),\sqrt{2\mu/\eta}\big)$. Letting for abbreviation purposes $t=\sqrt{2\mu/\eta}$, the question is to find a function $u(x)$ such that $$F\big(u(x),t\big)=\sqrt{\eta/2}~x$$ Taking the derivative (wrt $x$) on both sides, $$(\partial_u F)\big(u(x),t\big)~\cdot u'(x) =\sqrt{\eta/2}$$
Which boils down to solving the differential equation $$\frac{u'(x)}{u(x)\sqrt{t-u(x)}\sqrt{t+u(x)}}=\text{constant}$$
The solutions of which are given by some formula involving hyperbolic secant.