Square Root Simplifying and Solving the problem to X

Question

$e^y = \frac{x}{a} + \sqrt{( 1 + (\frac{x}{a})^2}$

The problem asks to solve this equation to x. My problem stands still by the square root.

Answer 1

If so we write $$e^y-\frac{x}{a}=\sqrt{1+\left(\frac{x}{a}\right)^2}$$ squaring we get $$e^{2y}+\left(\frac{x}{a}\right)^2-2e^y\times \frac{x}{a}=1+\left(\frac{x}{a}\right)^2$$ Combining like terms and isolating $x$ we get $$x=a\times \left(\frac{e^{2y}-1}{2e^y}\right)$$

Answer 2

Note that

• $\operatorname{arsinh} t = \ln\left(t+\sqrt{1+t^2}\right)$

Setting $t = \frac{x}{a}$, your equation becomes $$e^y = t+\sqrt{1+t^2} \Leftrightarrow y = \operatorname{arsinh} t$$

Hence,

$$t = \frac{x}{a} = \sinh y \Leftrightarrow \boxed{x= a\sinh y = a\frac{e^y-e^{-y}}{2}}$$