Solve dy=cos(tan^-1e^-x)x-1

$$d y = \cos ( \tan ^ { - 1 } e ^ { - x } ) x - 1$$

Solve for d

$$\left\{\begin{matrix}d=\frac{xe^{x}}{y\sqrt{e^{2x}+1}}-\frac{1}{y}\text{, }&y\neq 0\\d\in \mathrm{R}\text{, }&-\frac{x}{\sqrt{\frac{1}{e^{2x}}+1}}+1=0\text{ and }y=0\end{matrix}\right.$$

Steps for Solving Linear Equation

The equation is in standard form.

$$yd=x\cos(\arctan(\frac{1}{e^{x}}))-1$$

Divide both sides by y.

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

Dividing by y undoes the multiplication by y.

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

Divide \frac{x}{\sqrt{1+\frac{1}{e^{2x}}}}-1 by y.

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