Solve (1+e^frac{x{8}})dx+e^frac{x{7}}(1-x/y)dy=0

$$( 1 + e ^ { \frac { x } { 8 } } ) d x + e ^ { \frac { x } { 7 } } ( 1 - \frac { x } { y } ) d y = 0$$

Solve for d (complex solution)

$$\left\{\begin{matrix}d=0\text{, }&y\neq 0\\d\in \mathrm{C}\text{, }&y=x-xe^{-\frac{x}{56}}-xe^{-\frac{x}{7}}\text{ and }x-xe^{-\frac{x}{56}}-xe^{-\frac{x}{7}}\neq 0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by y.

$$\left(1+e^{\frac{x}{8}}\right)dxy+e^{\frac{x}{7}}\left(1-\frac{x}{y}\right)dyy=0$$

Multiply y and y to get y^{2}.

$$\left(1+e^{\frac{x}{8}}\right)dxy+e^{\frac{x}{7}}\left(1-\frac{x}{y}\right)dy^{2}=0$$

Use the distributive property to multiply 1+e^{\frac{x}{8}} by d.

$$\left(d+e^{\frac{x}{8}}d\right)xy+e^{\frac{x}{7}}\left(1-\frac{x}{y}\right)dy^{2}=0$$

Use the distributive property to multiply d+e^{\frac{x}{8}}d by x.

$$\left(dx+e^{\frac{x}{8}}dx\right)y+e^{\frac{x}{7}}\left(1-\frac{x}{y}\right)dy^{2}=0$$

Use the distributive property to multiply dx+e^{\frac{x}{8}}dx by y.

$$dxy+e^{\frac{x}{8}}dxy+e^{\frac{x}{7}}\left(1-\frac{x}{y}\right)dy^{2}=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{y}{y}.

$$dxy+e^{\frac{x}{8}}dxy+e^{\frac{x}{7}}\left(\frac{y}{y}-\frac{x}{y}\right)dy^{2}=0$$

Since \frac{y}{y} and \frac{x}{y} have the same denominator, subtract them by subtracting their numerators.

$$dxy+e^{\frac{x}{8}}dxy+e^{\frac{x}{7}}\times \frac{y-x}{y}dy^{2}=0$$

Express \frac{y-x}{y}d as a single fraction.

$$dxy+e^{\frac{x}{8}}dxy+e^{\frac{x}{7}}\times \frac{\left(y-x\right)d}{y}y^{2}=0$$

Express \frac{\left(y-x\right)d}{y}y^{2} as a single fraction.

$$dxy+e^{\frac{x}{8}}dxy+e^{\frac{x}{7}}\times \frac{\left(y-x\right)dy^{2}}{y}=0$$

Cancel out y in both numerator and denominator.

$$dxy+e^{\frac{x}{8}}dxy+e^{\frac{x}{7}}dy\left(-x+y\right)=0$$

Use the distributive property to multiply e^{\frac{x}{7}}dy by -x+y.

$$dxy+e^{\frac{x}{8}}dxy-e^{\frac{x}{7}}dyx+e^{\frac{x}{7}}dy^{2}=0$$

Combine all terms containing d.

$$\left(xy+e^{\frac{x}{8}}xy-e^{\frac{x}{7}}yx+e^{\frac{x}{7}}y^{2}\right)d=0$$

The equation is in standard form.

$$\left(xye^{\frac{x}{8}}-xye^{\frac{x}{7}}+xy+y^{2}e^{\frac{x}{7}}\right)d=0$$

Divide 0 by xy+e^{\frac{1}{8}x}xy-e^{\frac{1}{7}x}yx+e^{\frac{1}{7}x}y^{2}.

$$d=0$$

Solve for d

$$\left\{\begin{matrix}d=0\text{, }&y\neq 0\\d\in \mathrm{R}\text{, }&y=x-xe^{-\frac{x}{56}}-xe^{-\frac{x}{7}}\text{ and }x-xe^{-\frac{x}{56}}-xe^{-\frac{x}{7}}\neq 0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by y.

$$\left(1+e^{\frac{x}{8}}\right)dxy+e^{\frac{x}{7}}\left(1-\frac{x}{y}\right)dyy=0$$

Multiply y and y to get y^{2}.

$$\left(1+e^{\frac{x}{8}}\right)dxy+e^{\frac{x}{7}}\left(1-\frac{x}{y}\right)dy^{2}=0$$

Use the distributive property to multiply 1+e^{\frac{x}{8}} by d.

$$\left(d+e^{\frac{x}{8}}d\right)xy+e^{\frac{x}{7}}\left(1-\frac{x}{y}\right)dy^{2}=0$$

Use the distributive property to multiply d+e^{\frac{x}{8}}d by x.

$$\left(dx+e^{\frac{x}{8}}dx\right)y+e^{\frac{x}{7}}\left(1-\frac{x}{y}\right)dy^{2}=0$$

Use the distributive property to multiply dx+e^{\frac{x}{8}}dx by y.

$$dxy+e^{\frac{x}{8}}dxy+e^{\frac{x}{7}}\left(1-\frac{x}{y}\right)dy^{2}=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{y}{y}.

$$dxy+e^{\frac{x}{8}}dxy+e^{\frac{x}{7}}\left(\frac{y}{y}-\frac{x}{y}\right)dy^{2}=0$$

Since \frac{y}{y} and \frac{x}{y} have the same denominator, subtract them by subtracting their numerators.

$$dxy+e^{\frac{x}{8}}dxy+e^{\frac{x}{7}}\times \frac{y-x}{y}dy^{2}=0$$

Express \frac{y-x}{y}d as a single fraction.

$$dxy+e^{\frac{x}{8}}dxy+e^{\frac{x}{7}}\times \frac{\left(y-x\right)d}{y}y^{2}=0$$

Express \frac{\left(y-x\right)d}{y}y^{2} as a single fraction.

$$dxy+e^{\frac{x}{8}}dxy+e^{\frac{x}{7}}\times \frac{\left(y-x\right)dy^{2}}{y}=0$$

Cancel out y in both numerator and denominator.

$$dxy+e^{\frac{x}{8}}dxy+e^{\frac{x}{7}}dy\left(-x+y\right)=0$$

Use the distributive property to multiply e^{\frac{x}{7}}dy by -x+y.

$$dxy+e^{\frac{x}{8}}dxy-e^{\frac{x}{7}}dyx+e^{\frac{x}{7}}dy^{2}=0$$

Combine all terms containing d.

$$\left(xy+e^{\frac{x}{8}}xy-e^{\frac{x}{7}}yx+e^{\frac{x}{7}}y^{2}\right)d=0$$

The equation is in standard form.

$$\left(xye^{\frac{x}{8}}-xye^{\frac{x}{7}}+xy+y^{2}e^{\frac{x}{7}}\right)d=0$$

Divide 0 by xy+e^{\frac{1}{8}x}xy-e^{\frac{1}{7}x}yx+e^{\frac{1}{7}x}y^{2}.

$$d=0$$