Solve y_{t}sqrt{y^2-1}=ce^x

$$y _ { t } \sqrt { y ^ { 2 } - 1 } = c e ^ { x }$$

Solve for c (complex solution)

$$c=\frac{y_{t}\sqrt{y^{2}-1}}{e^{x}}$$

Steps for Solving Linear Equation

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

$$ce^{x}=y_{t}\sqrt{y^{2}-1}$$

The equation is in standard form.

$$e^{x}c=y_{t}\sqrt{y^{2}-1}$$

Divide both sides by e^{x}.

$$\frac{e^{x}c}{e^{x}}=\frac{y_{t}\sqrt{y^{2}-1}}{e^{x}}$$

Dividing by e^{x} undoes the multiplication by e^{x}.

$$c=\frac{y_{t}\sqrt{y^{2}-1}}{e^{x}}$$

Solve for c

$$c=\frac{y_{t}\sqrt{y^{2}-1}}{e^{x}}$$

$$|y|\geq 1$$

Steps for Solving Linear Equation

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

$$ce^{x}=y_{t}\sqrt{y^{2}-1}$$

The equation is in standard form.

$$e^{x}c=y_{t}\sqrt{y^{2}-1}$$

Divide both sides by e^{x}.

$$\frac{e^{x}c}{e^{x}}=\frac{y_{t}\sqrt{y^{2}-1}}{e^{x}}$$

Dividing by e^{x} undoes the multiplication by e^{x}.

$$c=\frac{y_{t}\sqrt{y^{2}-1}}{e^{x}}$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=\ln(\frac{y_{t}\sqrt{y^{2}-1}}{c})+2\pi n_{1}i\text{, }n_{1}\in \mathrm{Z}\text{, }&y\neq 1\text{ and }y\neq -1\text{ and }y_{t}\neq 0\text{ and }c\neq 0\\x\in \mathrm{C}\text{, }&\left(y_{t}=0\text{ or }y=-1\text{ or }y=1\right)\text{ and }c=0\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=\ln(\frac{y_{t}\sqrt{y^{2}-1}}{c})\text{, }&\left(y_{t}>0\text{ and }c>0\text{ and }|y|>1\right)\text{ or }\left(y_{t}<0\text{ and }c<0\text{ and }|y|>1\right)\\x\in \mathrm{R}\text{, }&|y|\geq 1\text{ and }\left(|y|=1\text{ or }y_{t}=0\right)\text{ and }c=0\end{matrix}\right.$$