Solve dy/sqrt{e^2y-1}=dx

$$\frac { d y } { \sqrt { e ^ { 2 y } } - 1 } = d x$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=-\frac{y}{1-e^{y}}\text{, }&\nexists n_{2}\in \mathrm{Z}\text{ : }y=i\times 2\pi n_{2}\\x\in \mathrm{C}\text{, }&d=0\text{ and }\nexists n_{1}\in \mathrm{Z}\text{ : }y=2\pi n_{1}i\end{matrix}\right.$$

Steps for Solving Linear Equation

Swap sides so that all variable terms are on the left hand side.

$$dx=\frac{dy}{\sqrt{e^{2y}}-1}$$

Divide both sides by d.

$$\frac{dx}{d}=\frac{dy}{\left(e^{y}-1\right)d}$$

Dividing by d undoes the multiplication by d.

$$x=\frac{dy}{\left(e^{y}-1\right)d}$$

Divide \frac{dy}{e^{y}-1} by d.

$$x=\frac{y}{e^{y}-1}$$

Solve for x

$$\left\{\begin{matrix}x=-\frac{y}{1-e^{y}}\text{, }&y\neq 0\\x\in \mathrm{R}\text{, }&d=0\text{ and }y\neq 0\end{matrix}\right.$$

Steps for Solving Linear Equation

Swap sides so that all variable terms are on the left hand side.

$$dx=\frac{dy}{\sqrt{e^{2y}}-1}$$

Divide both sides by d.

$$\frac{dx}{d}=\frac{dy}{\left(e^{y}-1\right)d}$$

Dividing by d undoes the multiplication by d.

$$x=\frac{dy}{\left(e^{y}-1\right)d}$$

Divide \frac{dy}{e^{y}-1} by d.

$$x=\frac{y}{e^{y}-1}$$

Solve for d (complex solution)

$$\left\{\begin{matrix}d=0\text{, }&\nexists n_{1}\in \mathrm{Z}\text{ : }y=2\pi n_{1}i\\d\in \mathrm{C}\text{, }&y+x-xe^{y}=0\text{ and }\nexists n_{1}\in \mathrm{Z}\text{ : }y=2i\pi n_{1}\end{matrix}\right.$$

Solve for d

$$\left\{\begin{matrix}d=0\text{, }&y\neq 0\\d\in \mathrm{R}\text{, }&x=-\frac{y}{1-e^{y}}\text{ and }y\neq 0\end{matrix}\right.$$