Solve (1+e^x/9)dx+(e^x/y)(1+x/y)dy=0

$$( 1 + e ^ { x / 9 } ) d x + ( e ^ { x / y } ) ( 1 + x / y ) d y = 0$$

Solve for d

$$\left\{\begin{matrix}d=0\text{, }&y\neq 0\\d\in \mathrm{R}\text{, }&x\left(e^{\frac{x}{9}}+1\right)+\left(x+y\right)e^{\frac{x}{y}}=0\text{ and }y\neq 0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by y.

$$\left(1+e^{\frac{x}{9}}\right)dxy+e^{\frac{x}{y}}\left(1+\frac{x}{y}\right)dyy=0$$

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

$$\left(1+e^{\frac{x}{9}}\right)dxy+e^{\frac{x}{y}}\left(1+\frac{x}{y}\right)dy^{2}=0$$

Use the distributive property to multiply 1+e^{\frac{x}{9}} by d.

$$\left(d+e^{\frac{x}{9}}d\right)xy+e^{\frac{x}{y}}\left(1+\frac{x}{y}\right)dy^{2}=0$$

Use the distributive property to multiply d+e^{\frac{x}{9}}d by x.

$$\left(dx+e^{\frac{x}{9}}dx\right)y+e^{\frac{x}{y}}\left(1+\frac{x}{y}\right)dy^{2}=0$$

Use the distributive property to multiply dx+e^{\frac{x}{9}}dx by y.

$$dxy+e^{\frac{x}{9}}dxy+e^{\frac{x}{y}}\left(1+\frac{x}{y}\right)dy^{2}=0$$

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

$$dxy+e^{\frac{x}{9}}dxy+e^{\frac{x}{y}}\left(\frac{y}{y}+\frac{x}{y}\right)dy^{2}=0$$

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

$$dxy+e^{\frac{x}{9}}dxy+e^{\frac{x}{y}}\times \frac{y+x}{y}dy^{2}=0$$

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

$$dxy+e^{\frac{x}{9}}dxy+e^{\frac{x}{y}}\times \frac{\left(y+x\right)d}{y}y^{2}=0$$

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

$$dxy+e^{\frac{x}{9}}dxy+e^{\frac{x}{y}}\times \frac{\left(y+x\right)dy^{2}}{y}=0$$

Cancel out y in both numerator and denominator.

$$dxy+e^{\frac{x}{9}}dxy+e^{\frac{x}{y}}dy\left(x+y\right)=0$$

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

$$dxy+e^{\frac{x}{9}}dxy+e^{\frac{x}{y}}dyx+e^{\frac{x}{y}}dy^{2}=0$$

Combine all terms containing d.

$$\left(xy+e^{\frac{x}{9}}xy+e^{\frac{x}{y}}yx+e^{\frac{x}{y}}y^{2}\right)d=0$$

The equation is in standard form.

$$\left(xye^{\frac{x}{9}}+xye^{\frac{x}{y}}+xy+y^{2}e^{\frac{x}{y}}\right)d=0$$

Divide 0 by xy+e^{\frac{1}{9}x}xy+e^{xy^{-1}}yx+e^{xy^{-1}}y^{2}.

$$d=0$$