Solve y=dy/dx-e^-xy(y+xdy/dx)/v=

$$y = \frac { d y } { d x } - \frac { e ^ { - x y } ( y + x \frac { d y } { d x } ) } { v } =$$

Solve for v

$$\left\{\begin{matrix}v=-\frac{1}{e^{xy}}\text{, }&y\neq 0\\v\neq 0\text{, }&y=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Variable v cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by v.

$$yv=v\frac{\mathrm{d}}{\mathrm{d}x}(y)-e^{\left(-x\right)y}\left(y+x\frac{\mathrm{d}}{\mathrm{d}x}(y)\right)$$

Use the distributive property to multiply e^{\left(-x\right)y} by y+x\frac{\mathrm{d}}{\mathrm{d}x}(y).

$$yv=v\frac{\mathrm{d}}{\mathrm{d}x}(y)-\left(e^{\left(-x\right)y}y+e^{\left(-x\right)y}x\frac{\mathrm{d}}{\mathrm{d}x}(y)\right)$$

To find the opposite of e^{\left(-x\right)y}y+e^{\left(-x\right)y}x\frac{\mathrm{d}}{\mathrm{d}x}(y), find the opposite of each term.

$$yv=v\frac{\mathrm{d}}{\mathrm{d}x}(y)-e^{\left(-x\right)y}y-e^{\left(-x\right)y}x\frac{\mathrm{d}}{\mathrm{d}x}(y)$$

Subtract v\frac{\mathrm{d}}{\mathrm{d}x}(y) from both sides.

$$yv-v\frac{\mathrm{d}}{\mathrm{d}x}(y)=-e^{\left(-x\right)y}y-e^{\left(-x\right)y}x\frac{\mathrm{d}}{\mathrm{d}x}(y)$$

Reorder the terms.

$$-v\frac{\mathrm{d}}{\mathrm{d}x}(y)+vy=-xe^{-xy}\frac{\mathrm{d}}{\mathrm{d}x}(y)-ye^{-xy}$$

Combine all terms containing v.

$$\left(-\frac{\mathrm{d}}{\mathrm{d}x}(y)+y\right)v=-xe^{-xy}\frac{\mathrm{d}}{\mathrm{d}x}(y)-ye^{-xy}$$

The equation is in standard form.

$$yv=-\frac{y}{e^{xy}}$$

Divide both sides by y.

$$\frac{yv}{y}=-\frac{\frac{y}{e^{xy}}}{y}$$

Dividing by y undoes the multiplication by y.

$$v=-\frac{\frac{y}{e^{xy}}}{y}$$

Divide -\frac{y}{e^{xy}} by y.

$$v=-\frac{1}{e^{xy}}$$

Variable v cannot be equal to 0.

$$v=-\frac{1}{e^{xy}}\text{, }v\neq 0$$

Solve for x

$$\left\{\begin{matrix}x=\frac{\ln(-\frac{1}{v})}{y}\text{, }&vy\neq 0\text{ and }v<0\\x\in \mathrm{R}\text{, }&y=0\text{ and }v\neq 0\end{matrix}\right.$$